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Register a new userOn the number of antipodal or strictly antipodal pairs of points in finite subsets of $mathbb{R}^d$, III
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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/2rceil $ 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.
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