The Fourier Transform and Firey Projections of Convex Bodies

In [F] Firey extended the notion of the Minkowski sum, and introduced, for each real p, a new linear combination of convex bodies, that he called p-sums. Lutwak [Lu2], [Lu3] showed that these Firey sums lead to a Brunn-Minkowski theory for each p ≥ 1. He introduced the notions of p-mixed volume, p-surface area measure, and proved an integral representation and inequalities for p-mixed volumes, including an analog of the Brunn-Minkowski inequality. As a result, he gave a solution of a generalization of the classical Minkowski problem. The Fourier analytic approach to sections and projections of convex bodies has recently been developed and has led to several results in the classical Brunn-Minkowski theory. In this paper we apply the Fourier analytic methods to study what we call Firey projections of convex bodies in the context of the Brunn-Minkowski-Firey theory. In particular we consider a generalization of Aleksandrov’s projection theorem and formulate and solve an analog of the Shephard problem for Firey projections. It was proved in [KRZ] that if the surface area measure of a convex body K is absolutely continuous, then