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Inequality with distances in finite metric space

Source: Miklos Schweitzer 2023, Problem 3

March 7, 2024
metric spaceinequalities

Problem Statement

Let X={x0,x1,…,xn}X =\{x_0, x_1,\ldots , x_n\} be the basis set of a finite metric space, where the points are inductively listed such that xkx_k maximizes the product of the distances from the points {x0,x1,…,xk−1}\{x_0, x_1,\ldots , x_{k-1}\} for each 1⩽k⩽n.1\leqslant k\leqslant n. Prove that if for each x∈Xx\in X we let Πx\Pi_x be the product of the distances from xx{} to every other point, then Πxn⩽2n−1Πx\Pi_{x_n}\leqslant 2^{n-1}\Pi_x for any x∈X.x\in X.
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