Orlicz centroid bodies. (English) Zbl 1206.49050

Lutwak, Erwin; Yang, Deane; Zhang, Gaoyong

J. Differ. Geom. 84, No. 2, 365-387 (2010).

Summary: The sharp affine isoperimetric inequality that bounds the volume of the centroid body of a star body (from below) by the volume of the star body itself is the Busemann-Petty centroid inequality. A decade ago, the \(L_p\) analogue of the classical Busemann-Petty centroid inequality was proved. Here, the definition of the centroid body is extended to an Orlicz centroid body of a star body, and the corresponding analogue of the Busemann-Petty centroid inequality is established for convex bodies.

MSC:

49Q20	Variational problems in a geometric measure-theoretic setting
52A40	Inequalities and extremum problems involving convexity in convex geometry

Keywords:

sharp affine isoperimetric inequality; centroid body of a star body; Busemann-Petty centroid inequality; Orlicz centroid body; convex bodies