Abstract
AbstractWe investigate equality cases in inequalities for Sylvester-type functionals. Namely, it was proven by Campi, Colesanti, and Gronchi that the quantityis 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 andWj is the j-th quermassintegral in ℝd, we prove that the functionalis 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).
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