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it is well known that in an @xmath2-tree the intersection of any two metric balls is itself a metric ball . in this paper we will show that this is actually a characterization of @xmath0-trees , and that , more generally , the geometry of the intersection of balls encodes information about the curvature of a geodesic metric space . recall from @xcite , @xcite , @xcite or @xcite that a geodesic metric space is _ hyperbolic _ ( in the sense of gromov ) if there is a constant @xmath3 such that for any geodesic triangle , any one side is contained in the @xmath4-neighborhood of the union of the two other sides . we prove the following characterization of hyperbolicity . [ qb]a geodesic metric space @xmath5 is hyperbolic if and only if the intersection of any two metric balls is at uniformly bounded hausdorff distance from a ball .    studying curvature in terms of the geometry of the intersection of metric balls turns out to be very natural and both gromov hyperbolicity and the notion of cat(@xmath1 ) geometry may be characterised in theses terms ( see section [ ext ] ) . tracking constants in the proof of theorem [ qb ] , one can show that the hyperbolicity constant depends only on the eccentricity constant . as pierre pansu pointed out to us it is a then an easy observation to deduce that the hyperbolicity bound varies linearly with the eccentricity bound so we obtain :    [ rtreeschar ] a geodesic metric is an @xmath0-tree if and only if the intersection of any two balls is a ball . this characterisation of @xmath2-trees was first conjectured in an early version of this paper and wenger has independently established the conjecture using very different methods , see @xcite . our approach is entirely self - contained . the following notion is crucial for our purposes . [ eccentricity]we say that a set @xmath6 has _ eccentricity less than _ @xmath4 ( for some @xmath3 ) if there is @xmath7 such that @xmath8 for some @xmath9 . by convention the eccentricity of the empty set is 0 . we shall see that the intersection of balls having uniformly bounded eccentricity is also equivalent to hyperbolicity ( proposition [ hyp=>qb ] and lemma [ qb=>hyp ] ) . the paper is organized as follows . in section [ bigons ] we discuss the geometry of @xmath10-quasigeodesics following an idea of papasoglu in @xcite and pomroy in @xcite , which is an important step in the proof . section [ div ] discusses divergence functions and a quantitative version of a theorem in @xcite and a classical argument implying hyperbolicity . section [ proofs ] collects the proofs of theorem [ qb ] and corollary [ rtreeschar ] . the idea is to show that hyperbolicity is equivalent to intersections of balls having uniformly bounded eccentricity . one technical difficulty lies in the fact that the centre and radius of a ball are not , in general , well defined . when they are an elementary proof can be given as in section [ ext ] . pomroy s work appeared in his warwick university masters dissertation but has never been published . in the appendix we take this opportunity to place his main theorem on the record with our own variation on the proof . the figures may be found at the back of the paper , inserted prior to the bibliography .    _ acknowledgements : _ the authors are extremely indebted to pierre pansu for pointing out to us that our bound on the hyperbolicity constant implied a linear bound via scaling . we would also like to thank theo buehler and deborah ruoss for pointing out an error in an early draft of this paper and chris hruska for the reference @xcite . finally we would like to thank hamish short for his comments on the classical argument proving that non - linear divergence of geodesics implies hyperbolicity . in this section we establish preliminary results concerning the geometry of geodesics and quasi - geodesics in a geodesic metric space . we start with the simple observation that if two geodesics are synchronously far apart , then they are asynchronously at least half as far apart as well . [ synchronous ] let @xmath11 and @xmath12 be geodesics with @xmath13 . if there exists @xmath14 such that @xmath15 then @xmath16 for all @xmath17 . suppose that there is an @xmath18 such that @xmath19 . then since @xmath11 is a geodesic we have @xmath20 . hence @xmath21 . but by hypothesis and the triangle inequality @xmath22 this is a contradiction . similarly , if there is an @xmath23 such that @xmath19 . then since @xmath11 is a geodesic we have @xmath24 . hence @xmath25 . but by hypothesis and the triangle inequality @xmath26 this is again a contradiction .    for a constant @xmath27 a @xmath10-quasigeodesic is a continuous map @xmath28\to x$ ] such that @xmath29 , @xmath30 and for all @xmath31 $ ] @xmath32 the points @xmath33 and @xmath34 are said to be the endpoints of @xmath11 . a @xmath10 quasi - geodesic bigon is a pair @xmath35 of @xmath10 quasi - geodesic bigons which have the same end points . the images of the quasigeodesics @xmath11 and @xmath12 in @xmath36 are called the _ sides _ of the @xmath37-bigon . for @xmath38 , we say that a @xmath37-bigon is @xmath39-_fat _ if there are @xmath40 such that @xmath41 and that it is @xmath39-_thin _ if it is not @xmath39-fat . the following remark provides an easy mechanism for constructing @xmath10 quasi - geodesics . let @xmath38 and @xmath42 in @xmath36 . a _ @xmath39-path _ from @xmath43 to @xmath44 is a continuous path @xmath45 from @xmath43 to @xmath44 such that , for any @xmath46 for some @xmath47 @xmath48 given any point on @xmath49 on a @xmath50-quasi - geodesic with end points @xmath42 we obtain a @xmath39-path by taking a broken geodesic @xmath51 . the main result in this section is that if two geodesics start at the same point and travel `` almost parallel '' for long enough , then this ensures the existence of @xmath52-bigons for any @xmath53 whose fatness depends on the distance between the two geodesics . the proof of this follows the outline in @xcite . more precisely we prove the following .    [ meat]let @xmath5 be a geodesic metric space , and choose integers @xmath54 with @xmath55 , @xmath56 . assume that there are two geodesics @xmath11 , @xmath12 and a real number @xmath57 such that @xmath29 and @xmath58 $ ] for all @xmath59 . if @xmath36 does not contain a @xmath60-fat @xmath61-bigon then @xmath62 in words , there is an upper bound on the length for which two geodesics can travel at a controlled distance .    for the purpose of the proof we introduce a few definitions . take @xmath63 $ ] , we call an element @xmath64 a @xmath65-_point _ if @xmath66 . we say that @xmath47 is an integral @xmath65-point if @xmath47 is a positive integer and a @xmath65-point . we define the @xmath65-_distance _ between two integral @xmath65-points @xmath67 by @xmath68 and set @xmath69 if and only if @xmath70 . note that since we are assuming that @xmath71 are integers @xmath72 . the interval @xmath73 $ ] is covered by @xmath74 disjoint half open intervals of the form @xmath75 , where @xmath76 $ ] .    given that @xmath36 does not contain a @xmath60-fat @xmath61-bigon we claim the following :    _ claim : _ for any two integral @xmath65-points @xmath47 and @xmath77 , if @xmath78 , then @xmath79 we prove the above claim by induction over @xmath80 : first , for @xmath81 we have @xmath70 and @xmath82 because @xmath70 is a @xmath65-point . so we assume the claim is true for @xmath83 and show it for @xmath84 . given integral @xmath65-points @xmath71 with @xmath85 we let @xmath86 be the integral @xmath65 points such that @xmath87 note that @xmath88 because @xmath89 , and hence both @xmath90 and @xmath91 are greater than or equal to @xmath92 . set @xmath93 , as shown in figure 1 . suppose for a contradiction that @xmath94 . we will show that this implies the broken geodesics @xmath95 and @xmath96 form a @xmath60-fat @xmath61-bigon . since @xmath36 contains no such bigons we will conclude that in fact @xmath94 .    to see that @xmath95 is a @xmath61-path we use the fact that @xmath97 and that , by our induction hypothesis , @xmath98 : @xmath99 the last inequality comes from our supposition that @xmath94 . a similar argument shows that the broken geodesic @xmath96 is also a @xmath61-path if @xmath94 . now let @xmath100 be a point on the path @xmath96 which minimises the distance to @xmath101 . in particular @xmath102 . if @xmath100 lies on the arc @xmath103 then , by the triangle inequality , we compute : @xmath104 it follows that @xmath105 , which is a contradiction since we assumed @xmath56 and @xmath106 . it follows that @xmath100 must lie on the arc @xmath107 . but applying lemma [ synchronous ] to the geodesics @xmath35 with @xmath108 we see that @xmath109 and hence the bigon is @xmath60-fat as required . hence @xmath110 completing the induction .    now by the triangle inequality @xmath111 hence @xmath112 , so @xmath113 . so , @xmath114 , and @xmath115 because there are at least @xmath116 integer points in @xmath117 and each of these is a @xmath65-point for one of the @xmath74 possible values of @xmath65 . recall that for a geodesic metric space @xmath5 , a _ divergence function _ is a map @xmath118 such that for all @xmath119 , all @xmath120 and all geodesics @xmath121 $ ] , @xmath122 $ ] such that @xmath123 , if @xmath124 is such that @xmath125 and @xmath126 is a path in the closure of @xmath127 from @xmath128 to @xmath129 , then the length of @xmath126 is at least @xmath130 . we say that _ geodesics diverge in @xmath36 _ if there is a divergence function @xmath131 so that @xmath132 . papasoglu showed in @xcite[corollary 1.3 ] that a geodesic metric space @xmath5 is hyperbolic if and only if geodesics diverge in @xmath36 . here we provide a quantitive version of this result in order to relate the hyperbolicity constant for a space to the eccentricity bound . first , following @xcite we provide candidates for a divergence function . let @xmath133 and for @xmath124 define @xmath134 since @xmath36 is a geodesic space any path joining two points @xmath135 must have length at least @xmath136 , and setting @xmath137 it is easy to see that the function @xmath138 is a divergence function for @xmath36 . [ thinbigonsquant]let @xmath5 be a metric space such that any @xmath37-bigon is @xmath139-thin . then for @xmath140 and @xmath141 , any @xmath142 such that @xmath143 satisfies @xmath144 in words , @xmath145 and hence geodesic diverge .    before starting with the proof we show an intermediate result . suppose that @xmath5 is a metric space such that @xmath37-bigons are @xmath146-thin for some constant @xmath147 and some @xmath53 . let @xmath148 and @xmath149 . if two geodesics @xmath11 and @xmath12 starting at the same point are such that @xmath150 for some @xmath7 and @xmath151 for some @xmath142 , then @xmath152\ ] ] for any @xmath153 $ ] . let us denote by @xmath154 . take @xmath155 to be the midpoint on a geodesic @xmath156 from @xmath128 to @xmath129 . then the broken geodesics @xmath157 and @xmath158 form the sides of an @xmath159-bigon , which by assumption is @xmath160-thin . according to lemma [ synchronous ] , this bigon is at least @xmath161-fat since the distance between the two geodesics is more than @xmath162 at time @xmath163 . hence @xmath164 . the upper bound is obtained in a similar way by looking at the broken geodesics @xmath165 and @xmath166 . they form a @xmath167-bigon , which by assumption is @xmath168-thin . but by lemma [ synchronous ] , this bigon is at least @xmath159-fat since the distance between the two geodesics is @xmath169 at time @xmath170 . hence @xmath171 as required . we now can easily prove proposition [ thinbigonsquant ] . since @xmath143 there are two geodesics @xmath11 and @xmath12 and @xmath120 such that    * @xmath29 * @xmath150 * @xmath172 .    we can apply theorem [ meat ] with @xmath173 , @xmath174 and @xmath175 which can be done using the previous lemma with @xmath176 . the assumptions on @xmath162 and @xmath177 ensure that @xmath178 .    from now on we assume that @xmath140 so that the function @xmath138 satisfies the conclusions of proposition [ thinbigonsquant ] for appropriate constants . the next step is to replace the divergence function @xmath138 by a divergence function @xmath179 of exponential growth . given a rectifyable path @xmath126 , let us denote by @xmath180 its length . for @xmath124 we set @xmath181 where @xmath35 are geodesics , @xmath182 and the infimum is taken over all geodesics @xmath35 and all points @xmath119 and all @xmath183 . it is clear that if we define @xmath184 then @xmath179 is a divergence function on @xmath36 and that @xmath185 for all @xmath186 . the following shows that this divergence function has exponential growth . [ exponentialdivergence]for any @xmath187 and @xmath188 , then @xmath189 where @xmath190 and @xmath191 . in particular , @xmath179 has exponential growth . we start with an intermediate result . [ smalllemma ] if @xmath11 and @xmath12 are geodesics with @xmath192 and @xmath57 satisfies @xmath150 then @xmath193 , where @xmath194 is as in proposition [ exponentialdivergence ] . let @xmath195 be a geodesic joining @xmath196 to @xmath197 . suppose for a contradiction that @xmath195 has length less than @xmath198 so no point on @xmath195 lies in the interior of the ball @xmath199 , where @xmath200 . concatenate @xmath195 with the terminal subarcs of @xmath35 of length @xmath198 to obtain a path joining @xmath201 to @xmath202 . no point on this path lies in the interior of the ball @xmath199 so it must have length at least @xmath203 . on the other hand we see that the path has length @xmath204 which by assumption is less than @xmath205 so we get @xmath206 . however @xmath207 by choice of @xmath194 and this is a contradiction . arguing by induction on @xmath208 it is enough to show that @xmath209 for any @xmath210 . if @xmath211 we are done so we can assume that @xmath212 is finite and therefore there are geodesics @xmath35 with @xmath182 and @xmath57 such that @xmath150 such that there is an arc @xmath126 in the closure of the complement of the ball @xmath213 which joins @xmath128 and @xmath129 and which has length less than @xmath214 . let @xmath215\mid \alpha(t)\in b(x , r+r+n)\}$ ] and @xmath216\mid \alpha(t)\in b(x , r+r+n)\}$ ] . note that @xmath217 and @xmath218 so the subarc @xmath219}$ ] must have length less than or equal to @xmath220 . let @xmath221 and @xmath222 be geodesics from @xmath43 to @xmath223 and @xmath224 respectively . by combining the triangle inequality lemma [ smalllemma ] we see that @xmath225 it follows that at least one of the three terms in the above sum must be greater than or equal to @xmath162 . it can not be the middle term for the following reason : if @xmath226 then , by definition of @xmath179 , any path joining @xmath227 to @xmath228 which lies in the closure of the complement of the ball @xmath229 must have length at least @xmath212 so in particular the subarc @xmath219}$ ] must have length at least @xmath212 . on the other hand we observed that this subarc has length at most @xmath220 so we see that @xmath230 . since @xmath231 we have that @xmath232 and since @xmath233 this is a contradiction . so we may assume ( interchanging @xmath11 and @xmath221 with @xmath12 and @xmath222 if necessary ) that @xmath234 . we construct a path @xmath235 in the complement of the ball @xmath236 joining @xmath128 to @xmath237 by concatenating the subarc @xmath238}$ ] with the subarc @xmath239 of @xmath221 joining @xmath237 to @xmath223 . since @xmath223 is within @xmath240 of @xmath43 we see that @xmath239 has length at most @xmath194 , and so @xmath235 has length at most @xmath241 . on the other hand the length of @xmath235 is bounded below by @xmath242 , so we see that @xmath243 but recall that @xmath231 so we see that @xmath244 . finally , since @xmath245 and by definition of @xmath246 , we have that @xmath247 . it follows that @xmath248 and so @xmath249    it follows from lemma [ exponentialdivergence ] that for any affine function @xmath250 there is some @xmath251 such that @xmath252 for all @xmath253 . the value @xmath251 depends only on the function @xmath138 and the constants @xmath254 and @xmath255 , though there may be no closed formula to compute it . it is clear that the value of @xmath251 may be bounded in terms of the values of the constants @xmath194 and @xmath246 appearing in lemma [ exponentialdivergence ] , and those depend only on the constant @xmath162 and the eccentricity bound @xmath52 as shown in the next lemma . this will enable us to show that there is an upper bound on the hyperbolicity constant for @xmath36 which is a function of @xmath52 alone . [ constants ] let @xmath256 be a geodesic metric space such that for each @xmath37 the @xmath37-bigons are @xmath257-thin . let @xmath140 , @xmath258 and @xmath259 as in proposition [ exponentialdivergence ] . then @xmath194 and @xmath260 are bounded above by functions of @xmath52 and @xmath162 . more precisely ,    1 . @xmath261 2 . @xmath262    \(1 ) apply proposition [ thinbigonsquant ] with @xmath263 . \(2 ) apply proposition [ thinbigonsquant ] with @xmath264 . [ thinbigons]let @xmath256 be a geodesic metric space such that for each @xmath37 the @xmath37-bigons are @xmath257-thin . then @xmath36 is @xmath265-hyperbolic , for some function @xmath4 depending on @xmath52 alone .    to ensure that @xmath140 we set @xmath266 let @xmath267 and choose geodesics @xmath268 , \alpha_x=[y , z ] , \alpha_y=[z , x]$ ] in @xmath36 . we denote by @xmath269 the reverse of the geodesic @xmath270 . we wish to estimate the thickness @xmath4 of this geodesic triangle @xmath271 using the exponential divergence function @xmath179 defined above . the following argument is adapted from that given in @xcite . let @xmath272 and set @xmath273 and @xmath274 . similarly we define @xmath275 , @xmath276 , @xmath277 , @xmath277 , @xmath278 and @xmath279 . now set @xmath280 , @xmath281 and @xmath282    case 1 . at least one of the values , say @xmath283 , is non - positive . in this case any point on @xmath284 is within @xmath162 of the other two sides , while any point on the subarcs @xmath285 or @xmath286 is within @xmath162 of the other two sides . there is @xmath287 on @xmath284 between @xmath277 and @xmath288 that is within @xmath162 of both @xmath289 and @xmath290 . hence the broken paths @xmath291 and @xmath292 form a @xmath162-bigon , consequently the triangle is @xmath293-thin . all three of the values @xmath294 are positive . in this case we will show that we can bound all three of these values by some uniform value @xmath295 given in terms of @xmath162 and @xmath52 , and hence , setting @xmath266 , @xmath295 depends on @xmath52 alone . once this is done it is clear that any point on any side of the triangle is within @xmath296 of some point on one of the other two sides , so it remains to find this bound @xmath295 . we can assume that @xmath297 . first note that if @xmath298 we can run the argument from case 1 with @xmath299 in place of @xmath162 , so we may assume that @xmath300 . we claim that the interior of @xmath301 does nt intersect the geodesic @xmath302 . indeed , since @xmath303 , the interior of the balls @xmath301 and @xmath304 have disjoint intersection . similarly , the balls @xmath301 and @xmath305 have disjoint interiors . but since @xmath306 , the arc @xmath302 is contained in the union of @xmath305 and @xmath304 .    now let @xmath307 and @xmath308 . the arcs @xmath309 , @xmath310 , @xmath311 , @xmath312 and @xmath313 are in the complement of the ball @xmath301 ( since they are either in the ball @xmath304 or @xmath305 ) . so concatenating those arcs we obtain a path in the closure of the complement of the ball @xmath314 of length @xmath315 . applying the divergence function to the geodesics @xmath316 emanating from the point @xmath43 we see that @xmath317 . now choose an integer @xmath318 so that @xmath319 $ ] where @xmath320 are the constants estimated in lemma [ constants ] . if @xmath321 then @xmath322 and , since @xmath246 , @xmath194 depend only on @xmath52 , we are done . if @xmath323 we can apply theorem [ exponentialdivergence ] to show that @xmath324 . it follows that @xmath208 can also be bounded above in terms of @xmath52 and @xmath162 . as noted before we may choose @xmath266 to obtain a bound on @xmath208 in terms of @xmath52 alone . we denote this bound by @xmath325 . this gives a bound on @xmath326 since @xmath327 . this in turn bounds the fatness of the triangle as less than or equal to @xmath328    we have shown that any geodesic triangle is either @xmath329-thin , which since @xmath266 means that the triangle is @xmath330-thin , or it is @xmath331-thin , or it is @xmath332-thin . in the second case and third case the constants @xmath333 can all be written in terms of @xmath52 alone . if we take @xmath334 then we see that the space @xmath5 is @xmath265-hyperbolic as required . the proof of theorem [ qb ] is a sequence of simple observations , combined with theorem [ thinbigons ] . recall that if @xmath335 are subsets of a metric space @xmath5 , then the hausdorff distance between @xmath336 and @xmath337 is given by @xmath338 where for @xmath186 , @xmath339 is the @xmath260-neighborhood of @xmath336 . it is not clear how having eccentricity less than a constant @xmath4 and being at hausdorff distance less than @xmath4 to a ball are related in general , but in case of intersection of balls in a metric space those notions are equivalent . our first observation in this section holds for any geodesic metric space and gives the interior radius of the intersection of two balls . [ general]let @xmath5 be a geodesic metric space . * for any @xmath340 with @xmath341 and @xmath342 , if the balls @xmath343 and @xmath344 are neither disjoint nor nested , then @xmath345 where @xmath346 and @xmath347 is any point on any geodesic between @xmath43 and @xmath44 , at distance @xmath348 from @xmath43 . * if @xmath349 and @xmath350 , then @xmath351 . \(1 ) if the balls @xmath343 and @xmath344 are neither disjoint nor nested , then we have @xmath352 and @xmath353 , and hence given any geodesic @xmath11 from @xmath43 to @xmath44 we may take a point @xmath347 on @xmath11 at distance @xmath348 from @xmath43 and a ball of radius @xmath260 around @xmath347 . then for @xmath354 @xmath355 and similarly @xmath356 . \(2 ) first notice that our assumptions on @xmath17 and @xmath47 being strictly smaller than @xmath357 show that @xmath43 and @xmath44 do not belong to @xmath358 . let @xmath359 and @xmath360 , so that @xmath361 , @xmath362 and @xmath363 . take a geodesic @xmath364 from @xmath43 to @xmath100 and a point @xmath365 on this geodesic , at distance @xmath163 from @xmath100 . such a point exists because @xmath43 does not belong to @xmath366 . since @xmath367 we have @xmath368 similarly , take a geodesic @xmath369 from @xmath44 to @xmath100 and a point @xmath287 on this geodesic , at distance @xmath163 from @xmath100 . since @xmath370 we have @xmath371 combining the 2 inequalities gives @xmath372 , hence @xmath373 .    we can now prove one implication of theorem [ qb ] and corollary [ rtreeschar ] . [ hyp=>qb ] if a geodesic metric space @xmath5 is @xmath4-hyperbolic with hyperbolicity constant less than or equal to @xmath3 , then both the eccentricity of the intersection of any two balls and the hausdorff distance from the intersection to a ball are both uniformly bounded by @xmath374 . take @xmath340 with @xmath375 and @xmath376 , with @xmath377 . we will show that the eccentricity of @xmath358 is less than @xmath374 . this implies the statement about hausdorff distance as well , by definition of hausdorff distance ( recalled in the beginning of this section ) .    according to lemma [ general ] part ( 1 ) either @xmath378 , @xmath379 , @xmath380 or @xmath381 , where @xmath347 and @xmath260 are as defined in that lemma . in the first three cases the eccentricity is clearly bounded by @xmath382 . in the remaining case it suffices to show that there is a constant @xmath52 independent of @xmath383 such that @xmath358 is contained in some ball of radius @xmath384 . we will show that in fact @xmath385 . now , for @xmath386 , let us estimate the distance to @xmath347 . since @xmath347 lies on a geodesic from @xmath43 to @xmath44 it is within @xmath4 of a point @xmath49 which lies on a geodesic from @xmath44 to @xmath365 or on a geodesic from @xmath43 to @xmath365 . we first assume that @xmath49 lies on a geodesic from @xmath43 to @xmath365 . by the triangle inequality , we have that @xmath387 . since @xmath49 lies on a geodesic from @xmath43 to @xmath365 this yields @xmath388 . now we have , as required : @xmath389 if @xmath49 lies on a geodesic from @xmath44 to @xmath365 instead , then we use the same argument switching the roles of @xmath42 and of @xmath40 . the following says that in a geodesic metric space such that the intersection of any two balls has uniformly bounded eccentricity , then the set of points on @xmath39-paths is uniformly close to a geodesic . [ dpathtravelclose]suppose that @xmath5 has a uniform bound @xmath53 on the eccentricity of the intersection of any two balls . then for any @xmath27 , given any two points @xmath340 , any point on a @xmath37-path from @xmath43 to @xmath44 is contained in the @xmath390-neighbourhood of any geodesic from @xmath43 to @xmath44 . let @xmath365 be a point on a @xmath37-path @xmath45 from @xmath43 to @xmath44 with @xmath391 , @xmath392 so that @xmath393 . if @xmath394 then @xmath395 and @xmath365 is within @xmath37 of @xmath44 . similarly if @xmath396 then @xmath397 and @xmath47 is within @xmath37 of @xmath43 so we may assume that both @xmath398 . now let @xmath399 , so that @xmath400 so by lemma [ general ] point ( 2 ) we see that any ball contained in @xmath401 has radius at most @xmath37 . it follows from the bounded eccentricity hypothesis that @xmath402 for some point @xmath403 . let @xmath11 be a geodesic from @xmath43 to @xmath44 and @xmath347 any point on @xmath11 at distance less than @xmath17 to @xmath43 and less than @xmath47 to @xmath44 ( such a point exists because @xmath404 ) . then @xmath405 , hence @xmath406 .    _ remark : taking @xmath407 in the lemma above shows that in a geodesic metric space such that the intersection of any two metric balls has eccentricity less than or equal to @xmath4 , any geodesic between two points is contained in a @xmath4-neighbourhood of any other geodesic between those two points . _    an analogous result holds in terms of hausdorff distance . [ dpathtravelclose2]suppose that @xmath5 has a uniform bound @xmath53 on the hausdorff distance from the intersection of any two balls to a ball . then for any @xmath27 , given any two points @xmath340 , any point on a @xmath37-path from @xmath43 to @xmath44 is contained in the @xmath408-neighbourhood of any geodesic from @xmath43 to @xmath44 . let @xmath365 be a point on a @xmath37-path @xmath45 from @xmath43 to @xmath44 with @xmath391 , @xmath392 so that @xmath393 . if @xmath409 then @xmath410 and @xmath365 is within @xmath411 of @xmath44 . similarly if @xmath412 then @xmath413 and @xmath47 is within @xmath411 of @xmath43 so we may assume that both @xmath414 . now let @xmath399 , so that @xmath400 . by assumption , @xmath415 , for some @xmath403 and some @xmath7 , which implies that @xmath416 and that @xmath417 . so by lemma [ general ] point ( 2 ) we see that any ball contained in @xmath418 has radius at most @xmath419 . it follows that @xmath420 . let @xmath11 be a geodesic from @xmath43 to @xmath44 and @xmath347 any point on @xmath11 at distance less than @xmath17 to @xmath43 and less than @xmath47 to @xmath44 ( such a point exists because @xmath404 ) . then @xmath421 , hence @xmath422 . we can now prove the other implication in theorem [ qb ] , namely that ( b ) implies ( a ) .    [ qb=>hyp]there is a function @xmath423 such that :    1 . if @xmath5 is a geodesic metric space with the property that the intersection of any two balls has eccentricity bounded by @xmath52 then @xmath36 is @xmath265-hyperbolic . if @xmath5 is a geodesic metric space with the property that the intersection of any two balls is at hausdorff distance less than @xmath424 then @xmath36 is @xmath265-hyperbolic . according to lemma [ dpathtravelclose ] or lemma [ dpathtravelclose2 ] , in such a metric space all @xmath10 bigons are @xmath139-slim . we conclude using theorem [ thinbigons ] above . as pierre pansu pointed out to us it immediately follows that we can take @xmath4 to be linear in @xmath52 .    for any @xmath425 the space @xmath5 is @xmath426 hyperbolic .    scaling the metric we see that @xmath427 has the property that the intersection of any two balls has eccentricity bounded by @xmath428 and so is @xmath429-hyperbolic . rescaling we see that @xmath256 is @xmath426-hyperbolic . we conclude this section with the proof of corollary 2 which asserts that @xmath0-trees are characterised by the property that the intersection of any two metric balls is a metric ball . one implication is given by proposition [ hyp=>qb ] . conversely , if the space @xmath5 has the property that the intersection of any two balls has eccentricity 0 then for any @xmath425 the intersection of any two balls has eccentricity bounded by @xmath52 and so the space is @xmath426 hyperbolic for all @xmath425 . it is therefore @xmath382-hyperbolic and hence must be an @xmath2-tree . _ remark : notice that in fact we do not need to assume that the intersection of any two balls is a ball to carry out the proof , only that the eccentricity of such an intersection is @xmath382 . a priori this is a weaker condition , however in an @xmath2-tree the intersection of two balls is always a ball and therefore , as a consequence of the theorem , the two conditions are equivalent . _ in theorem [ qb ] and corollary [ rtreeschar ] , the assumption that the metric space @xmath5 be geodesic might not be needed . the notion of hyperbolic spaces extends to non - geodesic metric spaces via the gromov product ( see e.g. dfinition 3 , page 27 of @xcite ) and it would be interesting to find an appropriate generalisation of these results to that context . in particular recall that a @xmath4-ultrametric space is a metric space @xmath5 which satisfies the following strengthened version of the triangle inequality , @xmath430 for all @xmath431 in @xmath36 . it is easy to see that at least two of @xmath432 , @xmath433 and @xmath434 differ by at most @xmath4 , meaning that any triangle is almost isoceles . these are examples of @xmath374-hyperbolic spaces in the sense of gromov , see @xcite section 1.2 on page 90 . it would be interesting to know if those spaces do satisfy the property that any intersection of two balls is almost a ball ( in some sense ) . we say that a geodesic metric space @xmath256 has the geodesic extension property if any geodesic arc @xmath435\rightarrow x$ ] extends to a geodesic @xmath436 , i.e. , @xmath437}=\gamma$ ] . the most important feature of a space with the geodesic extension property is that centres and radii of balls are well defined ( this is easily checked ) . more precisely we will use the following . [ outeradius]let @xmath5 be a space with the geodesic extension property . for any @xmath340 and @xmath342 , with @xmath438 . then @xmath439 is the biggest ball that fits in @xmath358 , where @xmath346 and @xmath347 is a point on any geodesic between @xmath43 and @xmath44 , at distance @xmath348 from @xmath43 . let @xmath399 . if @xmath440 then @xmath441 and there is nothing to prove . so let us assume that @xmath442 . take @xmath186 and @xmath443 such that @xmath444 . let @xmath445 and @xmath446 , so that @xmath447 . since @xmath448 , we deduce that @xmath449 , and similarly , since @xmath450 , we deduce that @xmath451 ( here we use the geodesic extension property ) . combining those two inequalities shows that @xmath452    if @xmath453 is the symmetric space of constant curvature @xmath454 there is a single triangle @xmath43 , @xmath44 , @xmath365 ( up to isometry ) with side lengths @xmath391 , @xmath392 and @xmath375 , so we define @xmath455 where @xmath347 is the point on the geodesic between @xmath43 and @xmath44 at distance @xmath456 from @xmath43 . ( this point exists because the triangle inequality ensures that @xmath457 so the distance is positive , and because @xmath458 implies that @xmath459 . ) we shall see that there is an analogue to theorem [ qb ] which characterises cat(@xmath1 ) geometry for @xmath454    [ cat0 ] given @xmath454 , a geodesic metric space @xmath5 with geodesic extension property is @xmath460 if and only if the eccentricity of the intersection of any two balls of respective radii @xmath17 and @xmath47 and at distance @xmath357 is bounded by @xmath461 . one implication is clear so we suppose by contradiction that the eccentricity of the intersection of any two balls of respective radii @xmath17 and @xmath47 and at distance @xmath357 is bounded by @xmath461 but that @xmath36 is not cat(@xmath1 ) . then there is a geodesic triangle @xmath462 in @xmath36 and a point @xmath463 on a geodesic @xmath11 from @xmath43 to @xmath44 such that @xmath464 . let @xmath465 where @xmath466 denotes the geodesic @xmath467 in the comparison triangle . similarly let @xmath468 . in other words , @xmath251 and @xmath469 are the nearest points left and right of @xmath49 that satisfy the cat(@xmath1 ) inequality . so for any @xmath470 in the open interval @xmath471 we have @xmath472 . notice that @xmath473 since the metric varies continuously with points . set @xmath474 , and @xmath475 . let @xmath476 .    now consider a geodesic triangle @xmath477 where the geodesic from @xmath478 to @xmath479 is taken to be the restriction of @xmath11 to the closed interval @xmath480 $ ] . clearly the geodesic triangle @xmath481 is a comparison triangle in @xmath482 . take the point @xmath483 on the above geodesic from @xmath478 to @xmath479 at distance @xmath484 from @xmath478 . ( this point exists because of the triangle inequality . ) first notice that @xmath483 has to be equal to either @xmath478 or @xmath479 , otherwise we would have @xmath485 and the intersection of @xmath486 with @xmath487 exceeds the allowed eccentricity . assume that @xmath488 ( if @xmath489 the argument is similar and we omit it ) . then @xmath490 so @xmath491 . it follows that the path given by concatenating the geodesics from @xmath365 to @xmath478 and from @xmath478 to @xmath479 is itself a geodesic . now let @xmath492 be the midpoint of the geodesic from @xmath478 to @xmath479 . this point is at distance @xmath493 from @xmath365 . inspecting the ( degenerate ) comparison triangle we see that @xmath494 is also at distance @xmath493 from @xmath495 but this contradicts our assumption that every point between @xmath478 and @xmath479 is further from @xmath365 than we see in the comparison triangle .    _ remark : following the arguments in the proof of theorem 1 we note that there is an alternative notion of a `` based '' eccentricity function which measures eccentricity from the defined centre @xmath347 for both the inscribed and circumscribed balls . in these terms it is easy to see that gromov hyperbolicity is equivalent to the existence of a uniform bound on the based eccentricity function , while the proof of theorem [ cat0 ] shows that cat(@xmath1 ) geometry is characterised by bounding the based eccentricity function in terms of the function @xmath496 . hence both notions of non - positive curvature may be naturally expressed in terms of eccentricity bounds . _ in @xcite papasoglu showed that for a graph hyperbolicity was equivalent to a bound on the thinness of geodesic bigons . as remarked before the same statement is not true for general geodesic metric spaces ( any non - hyperbolic cat(0 ) space furnishes a counter example since uniqueness fo geodesics gives a bound of @xmath382 on the fatness of geodesic bigons . ) the point is that the bound on the fatness of geodesic bigons in a graph gives an automatic bound on the fatness of @xmath497 quasi - geodesic bigons and papasoglu remarks that there is a natural generalisation of the result as follows . the theorem appears in the masters dissertation of pomroy @xcite but to the best of our knowledge no proof exists in the literature . we offer a proof of the result in order to place it on the record . we argue by contradiction , similarly to corollary [ thinbigons ] and using those particular divergence functions . according to theorem [ thinbigonsquant ] for any @xmath500 , the functions @xmath138 as defined above never tend to infinity , and hence there is @xmath501 so that @xmath502 . this means that for any @xmath503 , there are two geodesics @xmath11 and @xmath12 and @xmath120 such that      let us fix @xmath56 so that @xmath506 and again let @xmath507 and @xmath508 be points on a geodesic from @xmath509 to @xmath510 and at distance less than @xmath499 from each other ( we can choose @xmath511 ) . for @xmath512 , we see a @xmath499-bigon as follows : one side is a geodesic from @xmath33 to @xmath513 , and the other is a broken geodesic @xmath514 . so our assumptions say that it is @xmath52-thin , and hence @xmath515 . now let @xmath516 points on a geodesic from @xmath128 to @xmath129 and at distance less than @xmath499 from each other ( we can choose @xmath517 ) . again we construct a @xmath499-bigon as follows : one side is a geodesic from @xmath33 to @xmath518 , and the other is a broken geodesic @xmath519 . moreover , since @xmath520 , it means that @xmath521 , and hence @xmath522 , so that @xmath523 . this means that the geodesics @xmath11 and @xmath12 fulfill the assumptions of theorem [ meat ] for any @xmath162 big enough ( i.e. , @xmath162 so that @xmath524 , and hence , taking @xmath162 big enough ( i.e. , so that @xmath525 ) , theorem [ meat ] contradicts our assumption that @xmath499-bigons are uniformly @xmath52-thin . _ remark : this may seem close to lemma 7.2 in @xcite which states that if @xmath526-quasigeodesics stay uniformly close to any geodesic between the endpoints , then the space is hyperbolic . however the proof of papasoglu s or pomroy s result is considerably more elaborate . _ bridson and a. haefliger . _ metric spaces of non - positive curvature . _ grundlehren der mathematischen wissenschaften , * 319*. springer - verlag , berlin , 1999 . m. coornaert , t. delzant and a. papadopoulos . _ gomtrie et thorie des groupes . _ les groupes hyperboliques de gromov . lecture notes in mathematics , * 1441*. springer - verlag , berlin , 1990 . e. ghys , p. de la harpe ed . _ sur les groupes hyperboliques daprs mikhael gromov . _ papers from the swiss seminar on hyperbolic groups held in bern , 1988 . progress in mathematics * 83 * , birkhuser . m. gromov . _ hyperbolic groups . _ s. gersten , ed . , msri publications , * 4 * ( 1987 ) . h. masur and y. minsky . _ geometry of the complex of curves . i. hyperbolicity . _ invent . math . * 138 * ( 1999 ) , no . 1 , 103149 . p. papasoglu , _ strongly geodesically automatic groups are hyperbolic _ , invent . ( 1995 ) * 2 * pp 323334 . j. pomroy , master thesis , warvick . et al _ notes on word hyperbolic groups in _ group theory from a geometrical viewpoint _ ( e. ghys , a. haefliger , a. verjovsky , ed ) proc . ictp trieste 1990 , worldscientific , singapore , 1991 , 364 . s. wenger . _ on gromov hyperbolicity and a characterization of real trees _ , preprint
we show that a geodesic metric space is hyperbolic in the sense of gromov if and only if intersections of balls have bounded eccentricity . 
 in particular , @xmath0-trees are characterized among geodesic metric spaces by the property that the intersection of any two balls is always a ball . 
 both gromov hyperbolicity and cat(@xmath1 ) geometry can be characterised in terms of the geometry of the intersection of balls .