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

We improve our earlier upper bound on the numbers of antipodal pairs of points among $n$ points in ${\mathbb{R}}^3$, to $2n^2\mathrm{/}5+O(n^c)$, 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$\mathrm{.}Let${\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 ${\underset{‾}{sa}}_d^s(n)$ was known for $d\le 3$ and any $n$. Moreover, ${\underset{‾}{sa}}_d^s(n)=\lceil n\mathrm{/}2\rceil$ was known for $n\ge 2d$ even, and $n\ge 4d+1$ odd. We show ${\underset{‾}{sa}}_d^s(n)=2d$ for $2d+1 \le n \le 4d-1$odd,wedetermine${\underline{sa}}^s_d(n)$for$d=4$andany$n$, and prove ${\underset{‾}{sa}}_d^s(2d-1)=3(d-1)$. 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 ${\underset{‾}{sa}}_d^s(n)$ for them, which are of the same order of magnitude, namely $\mathrm{\Theta }((d-k)d)$. We present a simple example of a strictly antipodal set in ${\mathbb{R}}^d$, of cardinality const,$\cdot 1.5874…^d$\mathrm{.}Wegivesimpleproofsofthefollowingstatements:if$n$segmentsin${$\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.