Antipodality of spherical designs with odd harmonic indices

TL;DR

This work identifies the minimal size of a non-antipodal spherical design with odd harmonic indices {1,3,...,2m-1} as 2m+1, obtained by establishing a parallel lower bound in interval designs and transferring it via projections onto lines. Central to the method are Newton's identities connecting power sums and elementary symmetric polynomials, which yield symmetry constraints for odd-degree moments. The authors prove that any non-antipodal spherical T_m-design must have at least 2m+1 points and provide constructions achieving this bound, thereby proving optimality. They also develop and analyze weighted interval designs to show sharpness of bounds and discuss embedding techniques to realize minimal non-antipodal designs in higher dimensions.

Abstract

We determine the smallest size of a non-antipodal spherical design with harmonic indices $\{1,3,\dots,2m-1\}$ to be $2m+1$, where $m$ is a positive integer. This is achieved by proving an analogous result for interval designs.

Theorems & Definitions (32)

• Lemma 2.1
• Proposition 2.2
• proof
• Definition 3.1
• Proposition 3.2
• proof
• Remark 3.3
• Definition 3.4
• Lemma 3.5
• proof