Volume of mixed bodies

Lutwak, Erwin

doi:10.1090/S0002-9947-1986-0825717-3

Volume of mixed bodies
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Trans. Amer. Math. Soc. 294 (1986), 487-500 Request permission

Abstract:

By using inequalities obtained for the volume of mixed bodies and the Petty Projection Inequality, (sharp) isoperimetric inequalities are derived for the projection measures (Quermassintegrale) of a convex body. These projection measure inequalities, which involve mixed projection bodies (zonoids), are shown to be strengthened versions of the classical inequalities between the projection measures of a convex body. The inequality obtained for the volume of mixed bodies is also used to derive a form of the Brunn-Minkowski inequality involving mixed bodies. As an application, inequalities are given between the projection measures of convex bodies and the mixed projection integrals of the bodies.

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