On L_p Brunn-Minkowski type inequalities for a general class of functionals

2025/02/28 by Malagón, Lidia Gordo, Nicolás, Jesús Yepes
#26B15 #28A25 #52A39 #FOS: Mathematics #Functional Analysis (math.FA) #Metric Geometry (math.MG) #Primary 52A40 #Secondary 52A20

paper · doi:10.48550/arxiv.2503.00153

Abstract

In this work, the L_p version (for p> 1) of the dimensional Brunn-Minkowski inequality for the standard Gaussian measure γ_n(⋅) on ℝⁿ is shown. More precisely, we prove that for any 0-symmetric convex sets with nonempty interior, any p>1, and every λ∈ (0,1), γ_n((1-λ)⋅ K+_p λ⋅ L)^p/n \geqslant (1-λ) γ_n(K)^p/n + λγ_n(L)^p/n, with equality, for some λ∈ (0,1) and p>1, if and only if K=L. This result, recently established without the equality conditions by Hosle, Kolesnikov and Livshyts, by using a different and functional approach, turns out to be the L_p extension of a celebrated result for the Minkowski sum (that is, for p=1) by Eskenazis and Moschidis (2021) on a problem by Gardner and Zvavitch (2010). Moreover, an L_p Brunn-Minkowski type inequality is obtained for the classical Wills functional W(⋅) of convex bodies. These results are derived as a consequence of a more general approach, which provides us with other remarkable examples of functionals satisfying L_p Brunn-Minkowski type inequalities, such as different absolutely continuous measures with radially decreasing densities.

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