Affine Isoperimetric Inequalities Research Articles

Formula omitted] mixed geominimal surface area

This paper deals with Lp geominimal surface area and its extension to Lp mixed geominimal surface area. We give an integral formula of Lp geominimal surface area by the p-Petty body and introduce the concept of Lp mixed geominimal surface area which is a natural extension of Lp geominimal surface area. Some inequalities, such as, analogues of Alexandrov–Fenchel inequalities, Blaschke–Santaló inequalities, and affine isoperimetric inequalities for Lp mixed geominimal surface areas are obtained.

Centroid bodies and the convexity of area functionals

We introduce a new volume definition on normed vector spaces. We show that the induced $k$-area functionals are convex for all $k$. In the particular case $k = 2$, our theorem implies that Busemann’s 2-volume density is convex, which was recently shown by Burago-Ivanov. We also show how the new volume definition is related to the centroid body and prove some affine isoperimetric inequalities.

Centro–affine curvature flows on centrally symmetric convex curves

We consider two types of $p$-centro affine flows on smooth, centrally symmetric, closed convex planar curves, $p$-contracting, respectively, $p$-expanding. Here $p$ is an arbitrary real number greater than 1. We show that, under any $p$-contracting flow, the evolving curves shrink to a point in finite time and the only homothetic solutions of the flow are ellipses centered at the origin. Furthermore, the normalized curves with enclosed area $\pi$ converge, in the Hausdorff metric, to the unit circle modulo SL(2). As a $p$-expanding flow is, in a certain way, dual to a contracting one, we prove that, under any $p$-expanding flow, curves expand to infinity in finite time, while the only homothetic solutions of the flow are ellipses centered at the origin. If the curves are normalized as to enclose constant area $\pi$, they display the same asymptotic behavior as the first type flow and converge, in the Hausdorff metric, and up to SL(2) transformations, to the unit circle. At the end, we present a new proof of $p$-affine isoperimetric inequality, $p\geq 1$, for smooth, centrally symmetric convex bodies in $\mathbb{R}^2$.

Lp Geominimal Surface Areas and their Inequalities

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 Santal\'{o} style inequality. Cyclic inequalities are established to obtain the monotonicity of the $L_p$ geominimal surface areas. Comparison between the $L_p$ geominimal surface area and the $p$-surface area is also provided.

The Orlicz affine isoperimetric inequality

In this paper, the Orlicz-affine surface area is introduced. Isoperimetric inequalities for this new afine surface area are established. Mathematics subject classification (2010): Primary 52A20; Secondary 53A15.

On the monotone properties of general affine surfaces under the Steiner symmetrization

In this paper, we prove that, if functions (concave) $\phi$ and (convex) $\psi$ satisfy certain conditions, the $L_{\phi}$ affine surface area is monotone increasing, while the $L_{\psi}$ affine surface area is monotone decreasing under the Steiner symmetrization. Consequently, we can prove related affine isoperimetric inequalities, under certain conditions on $\phi$ and $\psi$, without assuming that the convex body involved has centroid (or the Santal\'{o} point) at the origin.

Cone-volume measures of polytopes

The cone-volume measure of a polytope with centroid at the origin is proved to satisfy the subspace concentration condition. As a consequence a conjectured (a dozen years ago) fundamental sharp affine isoperimetric inequality for the U-functional is completely established – along with its equality conditions.

The mixed $L_p$ geominimal surface areas for multiple 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, Santal\'{o} 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 for two convex bodies.

BRUNN–MINKOWSKI TYPE INEQUALITIES FOR Lp MOMENT BODIES

AbstractAbout 15 years ago, Lutwak and Zhang (E. Lutwak and G. Zhang, Blaschke–Santalo inequalities, J. Differ. Geom. 47 (1997), 1–16) introduced the notion of Lp moment bodies and established important volume inequalities for them, which were recently generalized by Haberl and Schuster (C. Haberl and E. Schuster, General Lp affine isoperimetric inequalities, J. Differ. Geom. 83 (2009), 1–26). In this paper, we establish new Brunn–Minkowski-type inequalities with respect to Blaschke Lp harmonic addition for the quermassintegrals and dual quermassintegrals of Lp moment bodies.

Volume inequalities for asymmetric Wulff shapes

Sharp reverse affine isoperimetric inequalities for asymmetric Wulff shapes and their polars are established, along with the characterization of all extremals. These new inequalities have as special cases previously obtained simplex inequalities by Ball, Barthe and Lutwak, Yang, and Zhang. In particular, they provide the solution to a problem by Zhang.

Affine Properties of Convex Equal-Area Polygons

In this paper we discuss some affine properties of convex equal-area polygons, which are convex polygons such that all triangles formed by three consecutive vertices have the same area. Besides being able to approximate closed convex smooth curves almost uniformly with respect to affine length, convex equal-area polygons admit natural definitions of the usual affine differential geometry concepts, like affine normal and affine curvature. These definitions lead to discrete analogous to the six-vertex theorem and an affine isoperimetric inequality. One can also define discrete counterparts of the affine evolute, parallels and the affine distance symmetry set preserving many of the properties valid for smooth curves.

Functional affine-isoperimetry and an inverse logarithmic Sobolev inequality

We give a functional version of the affine isoperimetric inequality for log-concave functions which may be interpreted as an inverse form of a logarithmic Sobolev inequality for entropy. A linearization of this inequality gives an inverse inequality to the Poincaré inequality for the Gaussian measure.

Inequalities for general mixed affine surface areas

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, Santal\'{o}-type inequalities, and affine isoperimetric inequalities for these general mixed affine surface areas.

On the L p affine isoperimetric inequalities

We obtain an isoperimetric inequality which estimate the affine invariant p-surface area measure on convex bodies. We also establish the reverse version of Lp-Petty projection inequality and an affine isoperimetric inequality of Γ − pK.

Relative entropy of cone measures and L p centroid bodies

Let $K$ be a convex body in $\mathbb R^n$. We introduce a new affine invariant, which we call $\Omega_K$, that can be found in three different ways: as a limit of normalized $L_p$-affine surface areas, as the relative entropy of the cone measure of $K$ and the cone measure of $K^\circ$, as the limit of the volume difference of $K$ and $L_p$-centroid bodies. We investigate properties of $\Omega_K$ and of related new invariant quantities. In particular, we show new affine isoperimetric inequalities and we show a "information inequality" for convex bodies.

Mini-Workshop: Valuations and Integral Geometry

As a generalization of the notion of measure, valuations have long played a central role in the integral geometry of convex sets. In recent years there has been a series of striking developments. Several examples were presented at this meeting, e.g. the work of Bernig and Fu on the integral geometry of groups acting transitively on the unit sphere, that of Hug and Schneider on kinematic and Crofton formulas for tensor valued valuations and a series of results by Ludwig and Reitzner on classifications of affine invariant notions of surface areas and of convex body valued valuations. Mathematics Subject Classification (2000): 52B45, 52A22, 53C65. Introduction by the Organisers The workshop Valuations and Integral Geometry, organized by Semyon Alesker, Andreas Bernig, and Franz Schuster was held from January 17th to January 23rd, 2010. The meeting was attended by 16 participants working in different areas such as convex and differential geometry or geometric measure theory. The program involved 3 lecture series by Fu, Ludwig, and Reitzner as well as several one hour and shorter lectures built around them. Some highlights of the program will be described in the following. In a 3 hour lecture series, Joseph Fu presented intriguing recent results about the product and convolution structures on the space of continuous translation invariant valuations. This new algebraic machinery has been the key tool for a fuller understanding of the kinematic formulas for groups acting transitively on the unit sphere obtained by Alesker, Bernig, and Fu. Fu described in detail 142 Oberwolfach Report 04/2010 the hermitian case and presented a mysterious conjecture concerning the integral geometry in complex space forms. In a related vein, Judit Abardia presented her joint work with Eduardo Gallego and Gil Solanes on the integral geometry in complex space forms. Some starting points for an integral geometry in Hermitian symmetric spaces were discussed by Hiroyuki Tasaki. Daniel Hug gave a very clear talk about kinematic formulas for tensor valuations which he recently obtained in a joint work with Rolf Schneider. Rolf Schneider talked about zonoids and Crofton formulas in Minkowski spaces, and Wolfgang Weil about translative kinematic formulas. Closely related to these developments is a generalization of the notion of valuations to smooth manifolds which was explained by Semyon Alesker in an impromptu evening lecture. On a different line of research, Monika Ludwig gave an extremely interesting 3 hour lecture series about her characterizations of convex body valued valuations compatible with affine transformations. These results are deeply connected with the theory of isoperimetric inequalities. Here, the valuation point of view has shed new light on some classical affine isoperimetric inequalities which were shown to hold for larger classes of valuations by Haberl and Schuster. These inequalities have led to new affine Lp Sobolev inequalities and an affine symmetrization principle presented in a one hour lecture by Christoph Haberl. Matthias Reitzner gave a 3 hour lecture series on his joint work with Monika Ludwig concerning their breakthrough in the characterization of upper semicontinuous SL(n) invariant valuations. Their results classified both of the classical SL(n) invariant notions of affine surface area affine surface area and centro-affine surface area which date back to Blaschke’s school of affine differential geometry. In fact, all of the Lp affine surface areas introduced by Lutwak in early 1990’s were completely characterized. The program of the workshop also involved several excellent talks by young researchers. Thomas Wannerer spoke about his extension of Ludwig’s characterization of the projection operator, one of the key concepts introduced by Minkowski for the study of projections of convex bodies. Andy Tsang gave a talk about his work on valuations defined on Lp function spaces, which presents a particularly exciting new area in the theory of valuations. Gautier Berck presented joint work with Juan-Carlos Alvarez on the use of Crofton formulas in the metrisability problem on Finsler manifolds. Gil Solanes spoke about Crofton and Gauss-Bonnet formulas which are invariant under the action of the Mobius group.

Inequalities for L p-mixed curvature images

Lutwak, Yang, and Zhang posed the notion of L p -curvature images and established several L p analogs of the affine isoperimetric inequality. In this article, the notion of L p -mixed curvature images is introduced, L p -curvature images being a special case. The properties and L p analogs of the affine isoperimetric inequality are established for L p -mixed curvature images.

Stability of the Blaschke–Santaló and the affine isoperimetric inequality

A stability version of the Blaschke–Santaló inequality and the affine isoperimetric inequality for convex bodies of dimension n ⩾ 3 is proved. The first step is the reduction to the case when the convex body is o-symmetric and has axial rotational symmetry. This step works for related inequalities compatible with Steiner symmetrization. Secondly, for these convex bodies, a stability version of the characterization of ellipsoids by the fact that each hyperplane section is centrally symmetric is established.

General affine surface areas

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.

Orlicz centroid bodies

The sharp affine isoperimetric inequality that bounds the volume of the centroid body of a star body (from below) by the volume of the star body itself is the Busemann-Petty centroid inequality. A decade ago, the $L_p$ analogue of the classical Busemann- Petty centroid inequality was proved. Here, the definition of the centroid body is extended to an Orlicz centroid body of a star body, and the corresponding analogue of the Busemann-Petty centroid inequality is established for convex bodies.