  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 $n+{d
\choose 2}-1$. Let $c_d(n)$ 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 $c_d(n)$ was
known for $d \le 3$ and any $n$. Moreover, $c_d(n) = lceil n/2\rceil$ was known
for $n \ge 2d$ even, and $n \ge 4d+1$ odd. We show $c_d(n) = 2d$ for $2d+1 \le
n \le 4d-1$ odd, and we determine $c_d(n)$ for $d=4$ and any $n$. The cases $d
\ge 5 $ and $d+2 \le n \le 2d-1$ remain open, but we give a lower and an upper
bound on $c_d(n)$ for them. 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.
