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摘要
We consider a functional on the space of convex bodies in R-n of the form F(K) = integral(Sn-1) f(u)Sn-1 (k,du), where f is an element of C(Sn-1) is a given continous function on the unit sphere of R (n) , K is a convex body in R-n , n %26gt;= 3, and S (n-1)(K,.) is the area measure of K. We prove that F satisfies an inequality of Brunn-Minkowski type if and only if f is the support function of a convex body, i.e., F is a mixed volume. As a consequence, we obtain a characterization of translation invariant, continuous valuations which are homogeneous of degree n-1 and satisfy a Brunn-Minkowski type inequality.
- 出版日期2014-4
