I believe that it is impossible that the length of all diagonals of a convex polygon with perimeter 1 is less than 1/4. It is easy to verify the case of quadrilateral: assume the two diagonals are of length $d_1,d_2$, then by trianglular inequality, we have $$2d_1+2d_2\ge 1,$$ so the length of the longer diagonal should be greater than 1/4. Is it true for other convex polygons?
2026-03-31 09:57:19.1774951039
Is it possible that the length of all diagonals of a convex polygon with perimeter 1 is less than 1/4?
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