The Lp-Busemann–Petty Centroid Inequality

The ratio between the volume of the p-centroid body of a convex body K in Rn and the volume of K attains its minimum value if and only if K is an origin symmetric ellipsoid. This result, the Lp-Busemann–Petty centroid inequality, was recently proved by Lutwak, Yang, and Zhang. In this paper we show that all the intrinsic volumes of the p-centroid body of K are convex functions of a time-like parameter when K is moved by shifting all the chords parallel to a fixed direction. The Lp-Busemann–Petty centroid inequality is a consequence of this general fact.