Let a convex quadrilateral ABCD with perimeter 1，d is the maximum of AB,AC,AD,BC,BD,CD,prove that d is not less than 1/3
 we can prove that parallelogram ABCD with perimeter 1，than one of AC,BD is more than 1/3
 but the general case is very difficult to solve.



the minimal diameter of a quadrilateral 
The answer given by ε-δ (a kite inscribed in a Reuleaux triangle) can be found in


Ball, D. G. (1973), "A generalisation of π", *Mathematical Gazette* 57 (402): 298–303, doi:[10.2307/3616052](http://dx.doi.org/10.2307%252F3616052), JSTOR [3616052](http://www.jstor.org/stable/3616052);


He doesn't give an explicit proof that this is optimal but says it can be done by "some tedious but not very difficult trigonometry. See also


Griffiths, David; Culpin, David (1975), "Pi-optimal polygons", *Mathematical Gazette* 59 (409): 165–175, doi:[10.2307/3617699](http://dx.doi.org/10.2307%252F3617699), JSTOR [3617699](http://www.jstor.org/stable/3617699)


for extensions to higher order polygons.


