The Lp-Busemann-Petty centroid inequality

The ratio between the volume of the p-centroid body of a convex body K in R-n and the volume of K attains its minimum value if and only if K is an origin symmetric ellipsoid. This result, the L-p-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 L-p-Busemann-Petty centroid inequality is a consequence of this general fact. (C) 2002 Elsevier Science (USA).