Mathieu Meyer, Elisabeth Werner
We give geometric interpretations of certain affine invariants of convex bodies. The affine invariants are the p-affine surface areas introduced by Lutwak. The geometric interpretations involve generalizations of the Santaló-bodies introduced by the authors in a previous paper.
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Several general mixed affine surface areas are introduced. We prove some important properties, such as, affine invariance, for these general mixed affine surface areas. We also establish new Alexandrov-Fenchel type inequalities, Santaló-type inequalities, and affine isoperimetric inequalities for these general mixed affine surface areas.
We prove new Alexandrov-Fenchel type inequalities and new affine isoperimetric inequalities for mixed $p$-affine surface areas. We introduce a new class of bodies, the illumination surface bodies, and establish some of their properties. We show, for instance, that they are not necessarily convex. We give geometric interpretations of $L_p$ affine surface areas, mixed $p$-affine surface areas and other functionals via these bodies. The surprising new element is that not necessarily convex bodies provide the tool for these interpretations.
In this paper, we prove that, if functions (concave) $ϕ$ and (convex) $ψ$ satisfy certain conditions, the $L_ϕ$ affine surface area is monotone increasing, while the $L_ψ$ affine surface area is monotone decreasing under the Steiner symmetrization. Consequently, we can prove related affine isoperimetric inequalities, under certain conditions on $ϕ$ and $ψ$, without assuming that the convex body involved has centroid (or the Santaló point) at the origin.
Extremal affine surface areas in a functional setting
本文在函数空间中引入极值仿射表面积，建立其对偶公式，证明其与函数积分幂成正比，从而推广并改进了凸体情形的相关结果。
A Characterization of Functional Affine Surface Areas
本文研究了凸Lipschitz函数空间上估值的特征，证明了此类估值可由常数项、体积和功能仿射表面积线性组合表示，并建立了有限值凸函数的对偶结果。
Let $K$ be a convex body in ${\bf R}^n$ and $B$ be the Euclidean unit ball in ${\bf R}^n$. We show that $$\mbox{lim}_{t\rightarrow 0} \frac{|K| -|K_t|}{|B| - |B_t|}= \frac{as(K)}{as(B)},$$ where $as(K)$ respectively $as(B)$ is the affine surface area of $K$ respectively $B$ and $\{K_t\}_{t\geq 0}$, $\{B_t\}_{t\geq 0}$ are general families of convex bodies constructed from $K$, $B$ satifying certain conditions. As a corollary we get results obtained in [M-W], [Schm],[S-W] and[W].
Weighted floating functions and weighted functional affine surface areas
本文提出与对数凹或$s$-凹函数相关的加权浮动函数，并据此定义加权泛函仿射表面积，阐明其与已有泛函及经典仿射表面积的联系。
We prove new $L_p$ affine isoperimetric inequalities for all $ p \in [-\infty,1)$. We establish, for all $p\neq -n$, a duality formula which shows that $L_p$ affine surface area of a convex body $K$ equals $L_\frac{n^2}{p}$ affine surface area of the polar body $K^\circ$.
Two families of general affine surface areas are introduced. Basic properties and affine isoperimetric inequalities for these new affine surface areas as well as for $L_ϕ$ affine surface areas are established.
The relationship between $L_p$ affine surface area and curvature measures is investigated. As a result, a new representation of the existing notion of $L_p$ affine surface area depending only on curvature measures is derived. Direct proofs of the equivalence between this new representation and those previously known are provided. The proofs show that the new representation is, in a sense, "polar" to that of Lutwak's and "dual" to that of Schütt & Werner's.
This paper aims to develop basic theory for the dual Orlicz $L_ϕ$ affine and geominimal surface areas for star bodies, which belong to the recent dual Orlicz-Brunn-Minkowski theory for star bodies. Basic properties for these new affine invariants will be provided. Moreover, related Orlicz affine isoperimetric inequality, cyclic inequality, Santaló style inequality and Alexander-Fenchel type inequality are established.
The Orlicz-Brunn-Minkowski theory receives considerable attention recently, and many results in the $L_p$-Brunn-Minkowski theory have been extended to their Orlicz counterparts. The aim of this paper is to develop Orlicz $L_ϕ$ affine and geominimal surface areas for single convex body as well as for multiple convex bodies, which generalize the $L_p$ (mixed) affine and geominimal surface areas -- fundamental concepts in the $L_p$-Brunn-Minkowski theory. Our extensions are different from the general affine surface areas by Ludwig (in Adv. Math. 224 (2010)). Moreover, our definitions for Orlicz $…
Let K be a convex body in $R^d$. A random polytope is the convex hull $[x_1,...,x_n]$ of finitely many points chosen at random in K. $\Bbb E(K,n)$ is the expectation of the volume of a random polytope of n randomly chosen points. I. Bárány showed that we have for convex bodies with $C^3$ boundary and everywhere positive curvature $$ c(d)\lim_{n \to \infty} \frac {vol_d(K)-\Bbb E(K,n)}{(\frac{vol_d(K)}{n})^{\frac{2}{d+1}}} =\int_{\partial K} κ(x)^{\frac{1}{d+1}}dμ(x) $$ where $κ(x)$ denotes the Gauß-Kronecker curvature. We show that the same formula holds for all convex bodies if $κ(x)$ denotes…
Some inequalities of isoperimetric type for the c-affine surface area
针对Schütt等人新引入的c-仿射表面积Ω^c，证明球体类中其由半径n/(n+1)的球最大化，建立Santaló型不等式，并得到涉及表面积的更复杂不等式。
In this paper, we develop a basic theory of Orlicz affine and geominimal surface areas for convex and $s$-concave functions. We prove some basic properties for these newly introduced functional affine invariants and establish related functional affine isoperimetric inequalities as well as functional Santaló type inequalities.
Affine Angles via Area Cross Ratio and Isoptic Hyperbolas
本文通过面积交叉比定义了仿射角，并证明其在保持选定方向的仿射变换下不变，同时揭示了等距双曲线与双曲幂之间的联系。
In this paper, we introduce the $L_p$ geominimal surface area for all $-n\neq p<1$, which extends the classical geominimal surface area ($p=1$) by Petty and the $L_p$ geominimal surface area by Lutwak ($p>1$). Our extension of the $L_p$ geominimal surface area is motivated by recent work on the extension of the $L_p$ affine surface area -- a fundamental object in (affine) convex geometry. We prove some properties for the $L_p$ geominimal surface area and its related inequalities, such as, the affine isoperimetric inequality and a Santaló style inequality. Cyclic inequalities are established to…
$L_p$-Integral Affine Surface Area-Variational Capacity Inequalities
针对$n\boldsymbol{\u2265}2$且$1<p<n$的情形，研究$L_p$积分仿射表面积与$L_p$变分容量的不等式问题，否定了Ludwig-Xiao-Zhang提出的两个相关猜想。
This paper is dedicated to the Orlicz-Petty bodies. We first propose the homogeneous Orlicz affine and geominimal surface areas, and establish their basic properties such as homogeneity, affine invariance and affine isoperimetric inequalities. We also prove that the homogeneous geominimal surface areas are continuous, under certain conditions, on the set of convex bodies in terms of the Hausdorff distance. Our proofs rely on the existence of the Orlicz-Petty bodies and the uniform boundedness of the Orlicz-Petty bodies of a convergent sequence of convex bodies. Similar results for the nonhomog…
In this article, we propose the notion of the general $p$-affine capacity and prove some basic properties for the general $p$-affine capacity, such as affine invariance and monotonicity. The newly proposed general $p$-affine capacity is compared with several classical geometric quantities, e.g., the volume, the $p$-variational capacity and the $p$-integral affine surface area. Consequently, several sharp geometric inequalities for the general $p$-affine capacity are obtained. These inequalities extend and strengthen many well-known (affine) isoperimetric and (affine) isocapacitary inequalities…
In this paper we further develop the theory of f-divergences for log-concave functions and their related inequalities. We establish Pinsker inequalities and new affine invariant entropy inequalities. We obtain new inequalities on functional affine surface area and lower and upper bounds for the Kullback-Leibler divergence in terms of functional affine surface area. The functional inequalities lead to new inequalities for L_p-affine surface areas for convex bodies.
In this paper, we introduce several mixed $L_p$ geominimal surface areas for multiple convex bodies for all $p\neq -n$. Our definitions are motivated from an equivalent formula for the mixed $p$-affine surface area. Some properties, such as the affine invariance, for these mixed $L_p$ geominimal surface areas are proved. Related inequalities, such as, Alexander-Fenchel type inequality, Santaló style inequality, affine isoperimetric inequalities, and cyclic inequalities are established. Moreover, we also study some properties and inequalities for the $i$-th mixed $L_p$ geominimal surface areas …
The $L_p$-floating area, curvature entropy, and isoperimetric inequalities on the sphere
该研究将欧氏几何中的$L_p$-仿射表面积等概念推广到球面空间，引入球面凸体的$L_p$-浮动面积与曲率熵，证明相关等周不等式、对偶公式与单调性，并推广到非负常曲率空间形式。
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