An Lp-Functional Busemann–Petty Centroid Inequality
Abstract For a convex body $K\subset \mathbb{R}^n$, let $\Gamma _pK$ be its $L_p$-centroid body. The $L_p$-Busemann–Petty centroid inequality states that $\operatorname{vol}(\Gamma _pK) \geq \operatorname{vol}(K)$, with equality if and only if $K$ is an ellipsoid centered at the origin. In this work, we prove inequalities for a type of functional $L_r$-mixed volume for $1 \leq r < n$ and establish, as a consequence, a functional version of the $L_p$-Busemann–Petty centroid inequality.
1999 ◽
Vol 51
(2)
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pp. 225-249
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Keyword(s):
Keyword(s):
1951 ◽
Vol 47
(1)
◽
pp. 245-247
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Keyword(s):