The functional inequality for the mixed quermassintegral

Chen, Fangwei; Fang, Junyong; Luo, Ming; Yang, Chen Ning

doi:10.1186/s13660-020-02521-7

Cited by 3 publications

(2 citation statements)

References 28 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…5]). More generally, mixed quermassintegrals and mixed integrals of functions were studied in [19] and [53].…”

Section: Background On Convex Functions and Log Concave Functionsmentioning

confidence: 99%

On Minkowski and Blaschke symmetrizations of functions and related applications

Hoehner¹

2023

Preprint

View full text Add to dashboard Cite

The Minkowski symmetral of an α-concave function is defined, and some of its fundamental properties are deduced. It is shown that almost all sequences of random Minkowski symmetrizations of a quasiconcave function converge in the L p metric (p ≥ 1) to a spherical decreasing mean width rearrangement. A sharp extended Urysohn's type inequality for quasiconcave functions is then derived.Using inner linearizations from convex optimization, an analogue of polytopes inscribed in a convex body is studied for log concave functions: Given a log concave function, we consider log-affine minorants defined in terms of the inner linearizations of the corresponding convex base function. As an application of the functional Minkowski symmetrizations, it is shown that any functional defined on the class of log concave functions which satisfies certain conditions is maximized by the reflectional hypo-symmetrization. As a corollary, it is shown that among all log concave functions of a given mean width, the reflectional hypo-symmetrization is hardest to approximate by "log-affine" minorants with a fixed number of break points. This result generalizes the classical fact which states that among all convex bodies of a given mean width, a Euclidean ball is hardest to approximate by inscribed polytopes with a fixed number of vertices.Finally, the Blaschke symmetral of a function is defined in a natural way, and some of its basic properties are proved. It is shown that under suitable assumptions, every quasiconcave function can be transformed into its symmetric decreasing surface area rearrangement by a sequence of Blaschke symmetrizations. Some applications of this result are also presented.

show abstract

“…5]). More generally, mixed quermassintegrals and mixed integrals of functions were studied in [19] and [53].…”

Section: Background On Convex Functions and Log Concave Functionsmentioning

confidence: 99%

On Minkowski and Blaschke symmetrizations of functions and related applications

Hoehner¹

2023

Preprint

View full text Add to dashboard Cite

show abstract

“…Therefore, the quantity W j (f ) can be interpreted in terms of the usual quermassintegrals of its super-level sets, which was originally considered in [13]. We remark that several works on quermassintegrals for functions have appeared in the literature, for example, see [13,17,18,44].…”

Section: Integral Representation Of L Ps Mixed Quermassintegral For F...mentioning

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 the framework of L summations for functions

Roysdon,

Xing

2023

Journal of Functional Analysis

View full text Add to dashboard Cite

The functional inequality for the mixed quermassintegral

Cited by 3 publications

References 28 publications

On Minkowski and Blaschke symmetrizations of functions and related applications

On Minkowski and Blaschke symmetrizations of functions and related applications

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

On the framework of L summations for functions

Contact Info

Product

Resources

About