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 real d-space. In particular, the maximum size of k-diametral point configurations of Minkowski d-spaces is obtained for given k>1 and d>1 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. This is a joint work with Zsolt Langi (Univ. Tech., Budapest).