  The structure of $k$-diametral point configurations in Minkowski $d$-space is
shown to be closely related to the properties of $k$-antipodal point
configurations in $\mathbb{R}^d$. In particular, the maximum size of
$k$-diametral point configurations of Minkowski $d$-spaces is obtained for
given $k\geq 2$ and $d\geq 2$ generalizing Petty's results (Proc. Am. Math.
Soc. 29: 369-374, 1971) on equilateral sets in Minkowski spaces. Furthermore,
bounds are derived for the maximum size of $k$-diametral point configurations
in Euclidean $d$-space. In the proofs convexity methods are combined with
volumetric estimates and combinatorial properties of diameter graphs.
