Characterizations of extremals for some functionals on convex bodies

We investigate equality cases in inequalities for Sylvester-type functionals. Namely, it was proven by Campi, Colesanti and Gronchi that the quantity ∫ x0∈K ... ∫ xn∈K [V (conv{x0, ..., xn})]dx0...dxn , n ≥ d, p ≥ 1 is maximized by triangles among all planar convex bodies K (parallelograms in the symmetric case). We show that these are the only maximizers, a fact proven by Giannopoulos for p = 1. Moreover, if h : R+ → R+ is a strictly increasing function and Wj is the j-th quermassintegral in Rd, we prove that the functional ∫ x0∈K0 ... ∫ xn∈Kn h(Wj(conv{x0, ..., xn}))dx0...dxn , n ≥ d is minimized among the (n + 1)-tuples of convex bodies of fixed volumes if and only if K0, ...,Kn are homothetic ellipsoids when j = 0 (extending a result of Groemer) and Euclidean balls with the same center when j > 0 (extending a result of Hartzoulaki and Paouris).