On the volume product of polygons
The volume product of a convex body in Euclidean space is the product of its volume and the volume of its polar body where the origin is chosen so that this is minimal. The Blaschke-Santaló inequality says that the volume product attains its maximum for ellipsoids. Here it is shown that in the plane the volume product of convex polygons with at most \(n\) vertices is attained by the affine images of regular polygons with \(n\) vertices. The authors also give new proofs of Mahler's inequalities which assert that lower bounds are attained by triangles or, in the centrally symmetric case, by parallelograms. The corresponding problem in dimensions greater than 2 is still unsolved, see, e.g. [\textit{R. Schneider}, Convex bodies: the Brunn-Minkowski theory. Cambridge: Cambridge University Press (1993; Zbl 0798.52001); Convex bodies: the Brunn-Minkowski theory. Paperback reprint of the hardback edition 1993, Cambridge: Cambridge University Press (2008; Zbl 1143.52002)] or \textit{K. J. Böröczky} and \textit{D. Hug} [Adv. Appl. Math. 44, No.~4, 309--328 (2010; Zbl 1191.52002)].
scientific article; zbMATH DE number 5968110
scientific article