On a Linear Refinement of the Prékopa-Leindler Inequality

Colesanti, Andrea; Gómez, Eugenia Saorín; Nicolás, Jesús Yepes

doi:10.4153/cjm-2015-016-6

Cited by 9 publications

(9 citation statements)

References 20 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Following the idea of the above proof but now using the linear refinement of the Prékopa-Leindler inequality (see [12,Theorem 1.5]), we immediately get the following improvement of (3.2) for the classical Wills functional.…”

Section: Bounding the Wills Functional Of A Convex Bodymentioning

confidence: 92%

“…Proof. Given two log-concave functions f, g : R n −→ R ≥0 (decaying to zero at infinity), if there exists a hyperplane H ∈ G(n, n − 1) such that P H f and P H g have the same (finite) integral, then the right-hand side in the Prékopa-Leindler inequality (2.6) (Theorem B for p = 0) can be replaced by the arithmetic mean of the integrals of f and g (see [12,Theorem 1.5]):…”

Section: Bounding the Wills Functional Of A Convex Bodymentioning

confidence: 99%

See 1 more Smart Citation

Further inequalities for the (generalized) Wills functional

Alonso–Gutiérrez

Cifre

Nicolás

2020

Commun. Contemp. Math.

View full text Add to dashboard Cite

The Wills functional W(K) of a convex body K, defined as the sum of its intrinsic volumes Vi(K), turns out to have many interesting applications and properties. In this paper we make profit of the fact that it can be represented as the integral of a log-concave function, which, furthermore, is the Asplund product of other two log-concave functions, and obtain new properties of the Wills functional (indeed, we will work in a more general setting). Among others, we get upper bounds for W(K) in terms of the volume of K, as well as Brunn-Minkowski and Rogers-Shephard type inequalities for this functional. We also show that the cube of edge-length 2 maximizes W(K) among all 0-symmetric convex bodies in John position, and we reprove the well-known McMullen inequality W(K) ≤ e V 1 (K) using a different approach.

show abstract

Section: Bounding the Wills Functional Of A Convex Bodymentioning

confidence: 92%

Section: Bounding the Wills Functional Of A Convex Bodymentioning

confidence: 99%

Further inequalities for the (generalized) Wills functional

Alonso–Gutiérrez

Cifre

Nicolás

2020

Commun. Contemp. Math.

View full text Add to dashboard Cite

show abstract

“…We notice that a positively decreasing function φ : R −→ R ≥0 is quasi-concave and furthermore φ(0) = sup x∈R φ(x). Thus, as a consequence of [11,Theorem 4.1] we get the following example, which will play a relevant role along this paper.…”

Section: Which Prevents Any Possible Brunn-minkowski Type Inequality ...mentioning

confidence: 92%

Brunn–Minkowski inequalities in product metric measure spaces

Ritoré

Nicolás

2018

Advances in Mathematics

View full text Add to dashboard Cite

Given one metric measure space X satisfying a linear Brunn-Minkowski inequality, and a second one Y satisfying a Brunn-Minkowski inequality with exponent p ≥ −1, we prove that the product X × Y with the standard product distance and measure satisfies a Brunn-Minkowski inequality of order 1/(1 + p −1 ) under mild conditions on the measures and the assumption that the distances are strictly intrinsic. The same result holds when we consider restricted classes of sets. We also prove that a linear Brunn-Minkowski inequality is obtained in X × Y when Y satisfies a Prékopa-Leindler inequality.In particular, we show that the classical Brunn-Minkowski inequality holds for any pair of weakly unconditional sets in R n (i.e., those containing the projection of every point in the set onto every coordinate subspace) when we consider the standard distance and the product measure of n one-dimensional real measures with positively decreasing densities. This yields an improvement of the class of sets satisfying the Gaussian Brunn-Minkowski inequality.Furthermore, associated isoperimetric inequalities as well as recently obtained Brunn-Minkowski's inequalities are derived from our results.

show abstract

“…By the monotonicity and α-concavity of Ω and the inclusion (23), this triple of functions satisfy h (C p,λ,t r γ + D p,λ,t r γ )…”

Section: A Novelmentioning

confidence: 99%

“…To understand the infimal convolution geometrically, we can see that the function u v whose epigraph is the Minkowski sum of the epigraphs of u and v [21][22][23]:…”

Section: Introductionmentioning

confidence: 99%

On the framework of $L_{p}$ summations for functions

Roysdon¹,

Xing²

2021

Preprint

View full text Add to dashboard Cite

We develop the framework of L p operations for functions by introducing two primary new types L p,s summations for p > 0: the L p,s convolution sum and the L p,s Asplund sum for functions. The first type is defined as the linear summations of functions in terms of the L p coefficients (C p,λ,t , D p,λ,t ), the so-called the L p,s supremal-convolution when p ≥ 1 and the L p,s inf-sup-convolution when 0 < p < 1, respectively. The second type L p,s summation is created by the L p averages of bases for s-concave functions. We show that they are equivalent in the case s = 0 (log-concave functions) and p ≥ 1. For the former type L p,s summation, we establish the corresponding L p -Borell-Brascamp-Lieb inequalities for all s ∈ [−∞, ∞] and p ≥ 1. Furthermore, in summarizing the conditions for these new types of L p -Borell-Brascamp-Lieb inequalities, we define a series of the L p,s concavity definitions for functions and measures. On the other hand, for the latter type L p,s Asplund summation, we discover the integral formula for L p,s mixed quermassintegral for functions via tackling the variation formula of quermassintegral of functions for p ≥ 1.

show abstract

On a Linear Refinement of the Prékopa-Leindler Inequality

Cited by 9 publications

References 20 publications

Further inequalities for the (generalized) Wills functional

Further inequalities for the (generalized) Wills functional

Brunn–Minkowski inequalities in product metric measure spaces

On the framework of $L_{p}$ summations for functions

Contact Info

Product

Resources

About