Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
8959477 | Journal of Functional Analysis | 2018 | 22 Pages |
Abstract
We showâ«Eâ©Î¸+f(x)dxâ«Ef(x)dxâ¥(kγ+1(n+1)γ+1)kγ+1γ for all k-dimensional subspaces EâRn, θâEâ©Snâ1, and all γ-concave functions f:Rnâ[0,â) with γ>0, 0<â«Rnf(x)dx<â, and â«Rnxf(x)dx at the origin oâRn. Here, θ+:={x:ãx,θãâ¥0}. As a consequence of this result, we get the following generalization of Grünbaum's inequality:volk(Kâ©Eâ©Î¸+)volk(Kâ©E)â¥(kn+1)k for all convex bodies KâRn with centroid at the origin, k-dimensional subspaces EâRn, and θâEâ©Snâ1. The lower bounds in both of our inequalities are the best possible, and we discuss the equality conditions.
Related Topics
Physical Sciences and Engineering
Mathematics
Algebra and Number Theory
Authors
S. Myroshnychenko, M. Stephen, N. Zhang,