The Ehrhard inequality

By C. Borell. 2003

105 citations recorded by the source corpus.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/fc0e83d04518b5d6e4d1a390e4c2e384a374043c

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In Discrete Mathematics and Combinatorics

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

The Minkowski problem in Gaussian probability space (2020, methodology): “See also Borell [6] for a characterization of log-concave measures.”
Pointwise symmetrization inequalities for Sobolev functions and applications (2009, background): “One way to deal with this problem is to use pointwise rearrangement inequalities; among the many contributions in this direction here we only mention just a few [56,118,117,70,9,20,54,3,38,35,81,82, 109,77,32], and refer the reader to the references therein.”
The dimensional Brunn–Minkowski inequality in Gauss space (2020, background): “Ehrhard’s original proof of (3) proceeded via Gaussian symmetrization and required both sets A and B to be convex, an assumption which was later removed by Borell in [Bor03] (see also [Lat96] for a partial result).”
An extension of Minkowski's theorem and its applications to questions about projections for measures (2016, methodology): “11 in the case of the standard Gaussian measure is possible to obtain using Ehrhard’s inequality (see Ehrhard [17], Borell [8]).”
On the Brunn-Minkowski inequality for general measures with applications to new isoperimetric-type inequalities (2015, background): “It turns out that, in this case, there is a very useful description of log-concave and s-concave measures due to Borell, see [B]: a measure µ is log-concave if and only if it has a density of the form ϕ = e−V , where V is convex (and may attain value +∞).”
Gaussian Brunn-Minkowski inequalities (2010, background): “Since the function Φ is (strictly) log concave (i.e., log Φ is (strictly) concave), Ehrhard’s inequality and its equality condition imply that for 0 t 1 and closed convex sets K and L in Rn, (6) γn ((1− t)K + tL) ≥ γn(K)1−tγn(L)t, with equality when γn(K)γn(L) > 0 if and only if K = L. Inequalit”
A general expression for the distribution of the maximum of a Gaussian field and the approximation of the tail (2006, result): “When the process is Gaussian, general inequalities allow to give bounds on 1 − FM(u) = P{M > u} as well as asymptotic results for u → +∞. A partial account of this well established theory, since the founding paper by Landau and Shepp [20] should contain - among a long list of contributors - the work”