Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
4667397 | Advances in Mathematics | 2008 | 23 Pages |
Abstract
Let f:Rn→R+ be a log-concave function and for z∈Rnz∈Rn, definefz(y)=infx∈Rne−〈x−z,y−z〉f(x)for every y∈Rn. We discuss the problem of finding a sharp lower bound to the productP(f)=infz∈Rn(∫Rnf(x)dx∫Rnfz(y)dy). We prove that if n=1n=1, then P(f)⩾eP(f)⩾e and characterize the case of equality. The same method allows to give a new simple proof of the fact that if f is sign-invariant, then for all n , P(f)⩾n4P(f)⩾4n. These inequalities are functional versions, with exact lower bounds, of the so-called inverse Santaló inequality for convex bodies, that we state and discuss as conjectures.
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Physical Sciences and Engineering
Mathematics
Mathematics (General)
Authors
M. Fradelizi, M. Meyer,