Logarithmically concave functions and sections of convex sets in $R^{n}$

Ball, Keith

doi:10.4064/sm-88-1-69-84

Cited by 231 publications

(285 citation statements)

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“…subspace, the curve r = r(θ) in S ⊥ such that r(θ) is the (n − 1)-dimensional volume of the intersection of K with the half-space S θ forms the boundary of a convex body in S ⊥ . Proved in this form by H. Busemann in 1949 and motivated by his theory of area in Finsler spaces, it is also important in geometric tomography (see [ The previous inequality is very closely related to one found earlier by Ball [8]. 18.11.…”

Section: S(k L ·) = S(k ·) + S(l ·)mentioning

confidence: 61%

The Brunn-Minkowski inequality

Gardner

2002

Bull. Amer. Math. Soc.

756

561

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Abstract. In 1978, Osserman [124] wrote an extensive survey on the isoperimetric inequality. The Brunn-Minkowski inequality can be proved in a page, yet quickly yields the classical isoperimetric inequality for important classes of subsets of R n , and deserves to be better known. This guide explains the relationship between the Brunn-Minkowski inequality and other inequalities in geometry and analysis, and some applications.

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Section: S(k L ·) = S(k ·) + S(l ·)mentioning

confidence: 61%

The Brunn-Minkowski inequality

Gardner

2002

Bull. Amer. Math. Soc.

756

561

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“…(5) We have seen above that, in the case when L is centrally symmetric about z, CI(L, z) = I(L, z) and Theorem 5 is nothing else but the classical Busemann's theorem [1]. Conversely, the following result holds.…”

Section: Remarks (1)mentioning

confidence: 76%

The Convex Intersection Body of a Convex Body

Meyer¹,

Reisner²

2011

Glasgow Math. J.

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Abstract. Let L be a convex body in ‫ޒ‬ n and z an interior point of L. We associate with L and z a new, convex and centrally symmetric, body CI (L, z). This generalizes the classical intersection body I(L, z) (whose radial function at u ∈ S n−1 is the volume of the hyperplane section of L through z, orthogonal to u). CI(L, z) coincides with I(L, z) if and only if L is centrally symmetric about z. We study the properties of CI(L, z).2010 Mathematics Subject Classification. 52A20.

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“…Since f is log-concave, it follows from Ball [3] that for every z ∈ R n , the set K z is a convex body. If we can prove that there exists z 0 ∈ R n such that the center of mass of K z0 is at the origin, we get the result from proposition 3 applied to f 1 (x) = f (x + z 0 ) and f 2 (x) = g(x + z 0 ).…”

Section: Proofmentioning

confidence: 99%

“…2) The idea of attaching a convex set of the form of K 1 to a log-concave function f 1 to prove a functional inequality was originally used by K. Ball in [3] and is also used by Klartag and Milman in [13].…”

Section: Remarks: 1)mentioning

confidence: 99%

Some functional forms of Blaschke–Santaló inequality

2006

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We establish new functional versions of the Blaschke-Santaló inequality on the volume product of a convex body which generalize to the nonsymmetric setting an inequality of K. Ball [2] and we give a simple proof of the case of equality. As a corollary, we get some inequalities for logconcave functions and Legendre transforms which extend the recent result of Artstein, Klartag and Milman [1], with its equality case.

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Logarithmically concave functions and sections of convex sets in $R^{n}$

Cited by 231 publications

References 0 publications

The Brunn-Minkowski inequality

The Brunn-Minkowski inequality

The Convex Intersection Body of a Convex Body

Some functional forms of Blaschke–Santaló inequality

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