Vol. 58, No. 2, 1975

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Dual mixed volumes

Erwin Lutwak

Vol. 58 (1975), No. 2, 531–538

A concept dual to the mixed volumes of Minkowski is introduced. Duals to the classical mixed volume inequalities of Minkowski, Fenchel and Aleksandrov are presented. As an application of this work a sharp isoperimetric inequality relating the mean width of a convex body and the cross-sectional measures of its polar body is obtained. This inequality implies that of all convex bodies of a given mean width the n-ball (centered at the origin) is the one whose polar body has minimal cross-sectional measures of any index. It further gives a sharp lower bound for the product of the mean widths of a convex body and its polar body.

Mathematical Subject Classification 2000
Primary: 52A40
Received: 4 January 1974
Revised: 25 March 1974
Published: 1 June 1975
Erwin Lutwak