scholarly journals The Minimal Perimeter of a Log-Concave Function

Mathematics ◽  
2020 ◽  
Vol 8 (8) ◽  
pp. 1365
Author(s):  
Niufa Fang ◽  
Zengle Zhang
Keyword(s):  
Extremal Problem ◽  
Isoperimetric Inequality ◽  
Sobolev Inequality ◽  
Convex Bodies ◽  
Concave Function ◽  
Concave Functions

Inspired by the equivalence between isoperimetric inequality and Sobolev inequality, we provide a new connection between geometry and analysis. We define the minimal perimeter of a log-concave function and establish a characteristic theorem of this extremal problem for log-concave functions analogous to convex bodies.

scholarly journals Functional Geominimal Surface Area and Its Related Affine Isoperimetric Inequality

2020 ◽  
Vol 2020 ◽  
pp. 1-8
Author(s):  
Niufa Fang ◽  
Jin Yang
Keyword(s):  
Surface Area ◽  
Isoperimetric Inequality ◽  
Convex Bodies ◽  
Mixed Volume ◽  
Concave Functions ◽  
Variation Formula ◽  
First Variation ◽  
The First Variation ◽  
Geominimal Surface Area ◽  
Affine Isoperimetric Inequality

The first variation of the total mass of log-concave functions was studied by Colesanti and Fragalà, which includes the Lp mixed volume of convex bodies. Using Colesanti and Fragalà’s first variation formula, we define the geominimal surface area for log-concave functions, and its related affine isoperimetric inequality is also established.

scholarly journals Further inequalities for the (generalized) Wills functional

2020 ◽  
pp. 2050011
Author(s):  
David Alonso-Gutiérrez ◽  
María A. Hernández Cifre ◽  
Jesús Yepes Nicolás
Keyword(s):  
Convex Body ◽  
Convex Bodies ◽  
General Setting ◽  
Concave Function ◽  
Upper Bounds ◽  
Edge Length ◽  
Symmetric Convex ◽  
Concave Functions ◽  
Intrinsic Volumes

The Wills functional [Formula: see text] of a convex body [Formula: see text], defined as the sum of its intrinsic volumes [Formula: see text], 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 [Formula: see text] in terms of the volume of [Formula: see text], as well as Brunn–Minkowski and Rogers–Shephard-type inequalities for this functional. We also show that the cube of edge-length 2 maximizes [Formula: see text] among all [Formula: see text]-symmetric convex bodies in John position, and we reprove the well-known McMullen’s inequality [Formula: see text] using a different approach.

scholarly journals On Mixed Quermassintegral for Log-Concave Functions

2020 ◽  
Vol 2020 ◽  
pp. 1-9
Author(s):  
Fangwei Chen ◽  
Jianbo Fang ◽  
Miao Luo ◽  
Congli Yang
Keyword(s):  
Convex Bodies ◽  
Concave Functions ◽  
Integral Expression

In this paper, the functional Quermassintegral of log-concave functions in ℝ n is discussed. We obtain the integral expression of the i th functional mixed Quermassintegral, which is similar to the integral expression of the i th mixed Quermassintegral of convex bodies.

An Isoperimetric Inequality for Isoptic Curves of Convex Bodies

2020 ◽  
Vol 75 (4) ◽  
Author(s):  
Jesús Jerónimo-Castro ◽  
Marco Antonio Rojas-Tapia ◽  
Ulises Velasco-García ◽  
Carlos Yee-Romero
Keyword(s):  
Isoperimetric Inequality ◽  
Convex Bodies

scholarly journals An isoperimetric inequality for uniformly log-concave measures and uniformly convex bodies

2008 ◽  
Vol 254 (5) ◽  
pp. 1235-1268 ◽  
Author(s):  
Emanuel Milman ◽  
Sasha Sodin
Keyword(s):  
Isoperimetric Inequality ◽  
Convex Bodies ◽  
Uniformly Convex

scholarly journals MATRIX SUBADDITIVITY INEQUALITIES AND BLOCK-MATRICES

2009 ◽  
Vol 20 (06) ◽  
pp. 679-691 ◽  
Author(s):  
JEAN-CHRISTOPHE BOURIN
Keyword(s):  
Positive Operator ◽  
Concave Function ◽  
Half Line ◽  
Concave Functions ◽  
Block Matrices ◽  
Open Questions ◽  
Normal Operators ◽  
Positive Half

We give a number of subadditivity results and conjectures for symmetric norms, matrices and block-matrices. Let A, B, Z be matrices of same size and suppose that A, B are normal and Z is expansive, i.e. Z*Z ≥ I. We conjecture that [Formula: see text] for all non-negative concave function f on [0,∞) and all symmetric norms ‖ · ‖ (in particular for all Schatten p-norms). This would extend known results for positive operator to all normal operators. We prove these inequalities in several cases and we propose some related open questions, both in the positive and normal cases. As nice applications of subadditivity results we get some unusual estimates for partitioned matrices. For instance, for all symmetric norms and 0 ≤ p ≤ 1, [Formula: see text] whenever the partitioned matrix is Hermitian or its entries are normal. We conjecture that this estimate for f(t) = tp remains true for all non-negative concave functions f on the positive half-line. Some results for general block-matrices are also given.

Uniform Kadec-Klee Lorentz Spaces Lw,1 and Uniformly Concave Functions

1996 ◽  
Vol 39 (3) ◽  
pp. 266-274 ◽  
Author(s):  
S. J. Dilworth ◽  
C. J. Lennard
Keyword(s):  
Recent Work ◽  
Concave Function ◽  
Lorentz Spaces ◽  
Sufficient Condition ◽  
Concave Functions ◽  
General Sufficient Condition ◽  
Weak Star

AbstractWe consider the notion of a uniformly concave function, using it to characterize those Lorentz spaces Lw,1 that have the weak-star uniform Kadec-Klee property as precisely those for which the antiderivative ϕ of w is uniformly concave; building on recent work of Dilworth and Hsu. We also derive a quite general sufficient condition for a twice-differentiable ϕ to be uniformly concave; and explore the extent to which this condition is necessary.

Convex bodies and concave functions

1995 ◽  
Vol 123 (2) ◽  
pp. 477-477
Author(s):  
M. Meyer ◽  
G. Mokobodzki ◽  
M. Rogalski
Keyword(s):  
Convex Bodies ◽  
Concave Functions

On characterizations of sausages via inequalities and roots of Steiner polynomials

2017 ◽  
Vol 17 (4) ◽  
Author(s):  
Jesús Yepes Nicolás
Keyword(s):  
Isoperimetric Inequality ◽  
Convex Bodies ◽  
Extremal Sets ◽  
Mean Width ◽  
The Family ◽  
Algebraic Properties ◽  
The Mean

AbstractWe prove that sausages are the family of ‘extremal sets’ in relation to certain linear improvements of Minkowski’s first inequality when working with projection/sections assumptions. In particular they characterize the equality cases of the corresponding linear refinements of both the isoperimetric inequality and Urysohn’s inequality. We also characterize sausages by algebraic properties of the roots of Steiner polynomials, in which other functionals of convex bodies such as the inradius, the mean width or the diameter are involved.

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