MathDB

Points with unit distance almost on a sphere

Source: Alibaba Global Math Competition 2021, Problem 5

July 4, 2021

Euclidean spacegeometrylengthcollege contests3D geometrysphere

Problem Statement

Suppose that

A

is a finite subset of

\mathbb{R}^d

such that
(a) every three distinct points in

A

contain two points that are exactly at unit distance apart, and
(b) the Euclidean norm of every point

v

A

satisfies

\sqrt{\frac{1}{2}-\frac{1}{2\vert A\vert}} \le \|v\| \le \sqrt{\frac{1}{2}+\frac{1}{2\vert A\vert}}.

Prove that the cardinality of

A

is at most

2d+4