# Strong Negative Type in Spheres

## TL;DR

This paper demonstrates that certain subsets of spheres with limited antipodal pairs exhibit strong negative type, enabling unique probability measure determination and consistent statistical tests based on expected distances.

## Contribution

It establishes that subsets of spheres with at most one antipodal pair have strong negative type, extending the understanding of negative type properties and their implications for statistical testing.

## Key findings

- Subsets with at most one antipodal pair have strong negative type.
- Expected distances uniquely determine probability measures on these sets.
- Distance covariance tests are consistent against all alternatives in such sets.

## Abstract

It is known that spheres have negative type, but only subsets with at most one pair of antipodal points have strict negative type. These are conditions on the (angular) distances within any finite subset of points. We show that subsets with at most one pair of antipodal points have strong negative type, a condition on every probability distribution of points. This implies that the function of expected distances to points determines uniquely the probability measure on such a set. It also implies that the distance covariance test for stochastic independence, introduced by Sz\'ekely, Rizzo and Bakirov, is consistent against all alternatives in such sets. Similarly, it allows tests of goodness of fit, equality of distributions, and hierarchical clustering with angular distances. We prove this by showing an analogue of the Cram\'er--Wold theorem.