MathDB

Let

X = \{(x,y) \in \mathbb{R}^2 | y \geq 0, x^2+y^2 = 1\} \cup \{(x,0),-1\leq x\leq 1\}

be the edge of the closed semicircle with radius 1.
a) Let

n>1

be an integer and

P_1,P_2,\dots,P_n \in X

. Show that there exists a permutation

\sigma \colon \{1,2,\dots,n\}\to \{1,2,\dots,n\}

such that

\sum_{j=1}^{n}|P_{\sigma(j+1)}-P_{\sigma(j)}|^2\leq 8

. Where

\sigma(n+1) = \sigma(1)

. b) Find all sets

\{P_1,P_2,\dots,P_n \} \subset X

such that for any permutation

\sigma \colon \{1,2,\dots,n\}\to \{1,2,\dots,n\}

\sum_{j=1}^{n}|P_{\sigma(j+1)}-P_{\sigma(j)}|^2 \geq 8

.
Where

\sigma(n+1) = \sigma(1)

Problem Statement