Maximizing the Sum of the Distances between Four Points on the Unit Hemisphere

TL;DR

This paper proves a geometric inequality for four points on a unit hemisphere, establishing the maximum sum of distances as 4+4*sqrt(2), using a combination of analytical and numerical methods.

Contribution

The paper introduces a novel proof technique combining local maximum analysis with numerical verification over small regions for a geometric inequality.

Findings

Maximum sum of distances is 4+4*sqrt(2).

Constructed a neighborhood around the local maximum for analysis.

Partitioned the feasible set into small cubes for numerical verification.

Abstract

In this paper, we prove a geometrical inequality which states that for any four points on a hemisphere with the unit radius, the largest sum of distances between the points is 4+4*sqrt(2). In our method, we have constructed a rectangular neighborhood of the local maximum point in the feasible set, which size is explicitly determined, and proved that (1): the objective function is bounded by a quadratic polynomial which takes the local maximum point as the unique critical point in the neighborhood, and (2): the rest part of the feasible set can be partitioned into a finite union of a large number of very small cubes so that on each small cube the conjecture can be verified by estimating the objective function with exact numerical computation.