Given a set of points $S$ on the Euclidean plane, *Onion Peeling* determines the nested set $H$ of convex hulls on $S$. Define an analytical formula on $S$ which produces a point, not necessarily in $S$, that falls inside the innermost convex hull in $H$. The formula must not use $H$.



Finding points inside innermost convex hull
To proceed further along the lines of Gerry Myerson's comment and Joseph O'Rourke's illustration and remarks, there is no real analytic formula to produce a point inside the innermost core as a function of $n$ points, when $n \ge 4$:


If you have a triangle with 1 point inside, then the only possible solution is that point.


If there were any analytic function, then it since it picks one of the vertices in an open neighborhood, it would need to pick that point globally. But that's impossible, since the configuration can be rearranged to make a different point the innermost one.


There is not even a continuous function that produces a point in the innermost hull, at least for 5 or more points; here the idea is illustrated with 9 points, so that the peeling the onion gives a nondegenerate inner hull:


[alt text http://dl.dropbox.com/u/5390048/OnionPeels.png](http://dl.dropbox.com/u/5390048/OnionPeels.png)


If 6 points are added to the edges of a triangle formed by 3 of the points, then an arbitrarily small perturbation makes the first peeling produce a quadrilateral as shown. 
The intersection of this quadrilateral with the two other analogous quadrilaterals is empty, so there can be no continuous choice near the degenerate configuraton.


For 5 points, 2 extra points can be added so that a perturbation makes one of them into the inner hull. For 6 points, it can be arranged to be not quite as degenerate, where the inner hulls become 3 line segments that have no common intersection.


