MathDB

Let

n

be a positive integer and let

T

denote the collection of points

(x_1, x_2, \ldots, x_n) \in \mathbb R^n

for which there exists a permutation

\sigma

1, 2, \ldots , n

such that

x_{\sigma(i)} - x_{\sigma(i+1) } \geq 1

for each

i=1, 2, \ldots , n.

Prove that there is a real number

d

satisfying the following condition: For every

(a_1, a_2, \ldots, a_n) \in \mathbb R^n

there exist points

(b_1, \ldots, b_n)

and

(c_1,\ldots, c_n)

T

such that, for each

i = 1, . . . , n,

a_i=\frac 12 (b_i+c_i) , |a_i - b_i| \leq d, \text{and} |a_i - c_i| \leq d.

Problem Statement