Cross 𝑖-sections of star bodies and dual mixed volumes

Lv, Songjun; Leng, Gangsong

doi:10.1090/S0002-9939-07-08997-6

Cross $i$-sections of star bodies and dual mixed volumes
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by Songjun Lv and Gangsong Leng PDF

Proc. Amer. Math. Soc. 135 (2007), 3367-3373 Request permission

Abstract:

In this paper, we establish an extension of Funk’s section theorem. Our result has the following corollary: If $K$ is a star body in $\mathbb {R}^n$ whose central $i$-slices have the same volume (with appropriate dimension) as the central $i$-slices of a centered body $M$, then the dual quermassintegrals satisfy $\widetilde {W}_j(M)\leq \widetilde {W}_j(K)$, for any $0\leq j<n-i$, with equality if and only if $K=M$. The case that $K$ is a centered body implies Funk’s section theorem.

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