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

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

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1. (i) Let S and T be two points on the boundary of Γ, with S lying on the side AB and T lying on the side PQ of Γ (See Fig).
Join TA, TB, TS.
ST lies between TA and TB in triangle TAB.
One of AST and BST is at least 90o, say AST90o.
ATTS.
But AT lies inside triangle APQ and one of ATP and ATQ is at least 90o, say ATP90o.
Then APAT.
TSATAP1
(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)
=2Z1Z22, 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.