On the number of antipodal or strictly antipodal pairs of points in finite subsets of $\mathbb{R}^d$, III

TL;DR

This paper improves bounds on the number of antipodal pairs in finite point sets in Euclidean spaces, providing exact values in certain cases, and establishing new bounds and examples for higher dimensions.

Contribution

It offers new upper bounds, exact minimal antipodal pair counts in specific dimensions, and constructs examples of strictly antipodal sets with exponential growth in dimension.

Findings

Improved upper bound on antipodal pairs in ${f R}^3$ to $2n^2/5 + O(n^c)$

Exact minimal antipodal pairs in convex position for ${f R}^d$

Characterization of strictly antipodal sets and bounds in various dimensions

Abstract

We improve our earlier upper bound on the numbers of antipodal pairs of points among $n$ points in ${\mathbb{R}}^3$, to $2n^2/5+O(n^c)$, for some $c<2$. We prove that the minimal number of antipodal pairs among $n$ points in convex position in ${\mathbb{R}}^d$, affinely spanning ${\mathbb{R}}^d$, is $n + d(d - 1)/2 - 1$. Let ${\underline{sa}}^s_d(n)$ be the minimum of the number of strictly antipodal pairs of points among any $n$ points in ${\mathbb{R}}^d$, with affine hull ${\mathbb{R}}^d$, and in strictly convex position. The value of ${\underline{sa}}^s_d(n)$ was known for $d \le 3$ and any $n$. Moreover, ${\underline{sa}}^s_d(n) = \lceil n/2\rceil $ was known for $n \ge 2d$ even, and $n \ge 4d+1$ odd. We show ${\underline{sa}}^s_d(n) = 2d$ for $2d+1 \le n \le 4d-1$ odd, we determine ${\underline{sa}}^s_d(n)$ for $d=4$ and any $n$, and prove ${\underline{sa}}^s_d(2d -1) = 3(d - 1)$. The cases $d \ge 5 $ and $d+2 \le n \le 2d - 2$ remain open, but we give a lower and an upper bound on ${\underline{sa}}^s_d(n)$ for them, which are of the same order of magnitude, namely $\Theta \left( (d-k)d \right) $. We present a simple example of a strictly antipodal set in ${\mathbb{R}}^d$, of cardinality const\,$\cdot 1.5874...^d$. We give simple proofs of the following statements: if $n$ segments in ${\mathbb{R}}^3$ are pairwise antipodal, or strictly antipodal, then $n \le 4$, or $n \le 3$, respectively, and these are sharp. We describe also the cases of equality.