Gaussian Brunn-Minkowski inequalities

By Richard J. Gardner, Artem Zvavitch. 2010

105 citations recorded by the source corpus.

A detailed investigation is undertaken into Brunn-Minkowski-type inequalities for Gauss measure. A Gaussian dual Brunn-Minkowski inequality is proved, together with precise equality conditions, and shown to be best possible from several points of view. A possible new Gaussian Brunn-Minkowski inequality is proposed, and proved to be true in some special cases. Throughout the study attention is paid to precise equality conditions and conditions on the coecients of dilatation. Interesting links are found to the S-inequality and the (B) conjecture. An example is given to show that convexity is needed in the (B) conjecture.

Subfields: Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/4a59234b8962580e00053d74da200211129fb325

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In Applied Mathematics

Citations in the corpus

The Minkowski problem in Gaussian probability space (2020, background): “Gardner-Zvavitch [25] observed that this inequality neither implies nor is implied by Ehrhard’s inequality.”
The dual Orlicz–Brunn–Minkowski theory ☆ (2014, background): “ Minkowski geometry, the local theory of Banach spaces, geometric tomography, and stereology; see [13] and the references given there. The literature is large and continues to grow. See, for example, [1, 6, 10, 14, 15, 19, 24, 32]. 2010 Mathematics Subject Classiﬁcation. Primary: 52A20, 52A30; secon”
Local Lp-Brunn-Minkowski inequalities for p<1 (2017, background): “). • The log-Brunn–Minkowski conjecture has been shown by Saroglou [58, 59] to be intimately related to the generalized B-conjecture. Further connections to a conjecture of R. Gardner and A. Zvavitch [25] on a dimensional Brunn–Minkowski inequality for even log-concave measures were observed by Livs”
Local $L^p$-Brunn-Minkowski inequalities for $p<1$ (2017, methodology): “Zvavitch [22] on a dimensional Brunn– Minkowski inequality for even log-concave measures were observed by Livshyts, Marsiglietti, P.”
The dimensional Brunn–Minkowski inequality in Gauss space (2020, background): “Since the formulation of the problem in [GZ10], there have been several partial results towards the general statement of Theorem 1.”
Representation formula for the entropy and functional inequalities (2010, background): “Since then a number of nice alternate proofs have been discovered, see [3,9,10] and the survey article [1].”
More on logarithmic sums of convex bodies (2014, background): “1] (see also [19]) immediately imply the following: Corollary 3.”
On the Brunn-Minkowski inequality for general measures with applications to new isoperimetric-type inequalities (2015, background): “However, we have ψ(a) = √ 2a2 + 2 for a ∈ [1, 2], which is strictly convex.”