Hi, I have a question and I hope anyone could answer it:

A convex polygon Γ is such that the distance between any two vertices of Γ does not exceed 1.

(i) Prove that the distance between any two points on the boundary of Γ does not exceed 1.

(ii) If X and Y are two distinct points inside Γ .Prove that there exists a point Z o the boundary of Γ such that XZ+YZ ≤ 1

Join TA, TB, TS.

ST lies between TA and TB in triangle TAB.

One of ∠AST and ∠BST is at least 90o, say ∠AST≥90o.

∴AT≥TS.

But AT lies inside triangle APQ and one of ∠ATP and ∠ATQ is at least 90o, say ∠ATP≥90o.

Then AP≥AT.

∴TS≤AT≤AP≤1

(ii) Let X and Y be points in the interior Γ.

Join XY and produce them on either side to meet the sides CD and EF of Γ at Z1 and Z2 respectively.

We have, (XZ1+YZ1)+(XZ2+YZ2)=(XZ1+XZ2)+(YZ1+YZ2)

=2Z1Z2≤2, by the first part.

Therefore one of the sums XZ1+YZ1 and XZ2+YZ2 is at most 1.

Z can be chosen accordingly as Z1 or Z2.