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.

Abstract. The volume of the polar body of a symmetric convex set K of R d is investigated. It is shown that its reciprocal is a convex function of the time t along movements, in which every point of K moves with constant speed parallel to a fixed direction.This result is applied to find reverse forms of the L p -Blaschke-Santaló inequality for two-dimensional convex sets.

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