On the improvement of concavity of convex measures

Marsiglietti, Arnaud

doi:10.1090/proc/12694

Cited by 19 publications

(17 citation statements)

References 17 publications

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“…As was shown in [27] (see also the third named author [28]), this statement follows from log-Brunn-Minkowski conjecture in the case of log-concave spherically invariant measures and when K and L are Euclidean balls. The latter is indeed true: it follows from the results of [15] and [36].…”

Section: Proof Of Theorem 13supporting

confidence: 68%

On the stability of Brunn–Minkowski type inequalities

Colesanti

Livshyts

Marsiglietti

2017

Journal of Functional Analysis

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Section: Proof Of Theorem 13supporting

confidence: 68%

On the stability of Brunn–Minkowski type inequalities

Colesanti

Livshyts

Marsiglietti

2017

Journal of Functional Analysis

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“…In [22] the second named author proved that the assertion of the B-conjecture for a measure µ with a radially decreasing density and a symmetric convex body K formally implies the 1/n-concavity of the measure µ on the set of dilates of K.…”

Section: Introductionmentioning

confidence: 99%

On the Brunn-Minkowski inequality for general measures with applications to new isoperimetric-type inequalities

Livshyts

Marsiglietti

Nayar

et al. 2017

Trans. Amer. Math. Soc.

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In this paper we present new versions of the classical Brunn-Minkowski inequality for different classes of measures and sets. We show that the inequalityholds true for an unconditional product measure µ with decreasing density and a pair of unconditional convex bodies A, B ⊂ R n . We also show that the above inequality is true for any unconditional logconcave measure µ and unconditional convex bodies A, B ⊂ R n . Finally, we prove that the inequality is true for a symmetric log-concave measure µ and a pair of symmetric convex sets A, B ⊂ R 2 , which, in particular, settles two-dimensional case of the conjecture for Gaussian measure proposed in [13].In addition, we deduce the 1/n-concavity of the parallel volume t → µ(A + tB), Brunn's type theorem and certain analogues of Minkowski first inequality.

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“…Despite the relevance of the proven inequalities, this work seems not to be so well known in the literature. We also refer the interested reader to the papers [12,18] for related topics involving inequalities for functions and measures.…”

Section: Proof Of Main Resultsmentioning

confidence: 99%

On a Linear Refinement of the Prékopa-Leindler Inequality

Colesanti¹,

Gómez²,

Nicolás³

2016

Can. j. math.

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Abstract. If f , g : R n −→ R ≥0 are non-negative measurable functions, then the Prékopa-Leindler inequality asserts that the integral of the Asplund sum (provided that it is measurable) is greater or equal than the 0-mean of the integrals of f and g. In this paper we prove that under the sole assumption that f and g have a common projection onto a hyperplane, the Prékopa-Leindler inequality admits a linear refinement. Moreover, the same inequality can be obtained when assuming that both projections (not necessarily equal as functions) have the same integral. An analogous approach may be also carried out for the so-called Borell-Brascamp-Lieb inequality.

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On the improvement of concavity of convex measures

Abstract: We prove that a general class of measures, which includes logconcave measures, is 1 n -concave according to the terminology of Borell, with additional assumptions on the measures or on the sets, such as symmetries. This generalizes results of Gardner and Zvavitch [8].

Cited by 19 publications

References 17 publications

On the stability of Brunn–Minkowski type inequalities

On the stability of Brunn–Minkowski type inequalities

On the Brunn-Minkowski inequality for general measures with applications to new isoperimetric-type inequalities

On a Linear Refinement of the Prékopa-Leindler Inequality

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