Affine Isoperimetric Inequalities Research Articles

Affine isoperimetric inequalities on flag manifolds

Abstract Building on work of Furstenberg and Tzkoni, we introduce $\mathbf {r}$ -flag affine quermassintegrals and their dual versions. These quantities generalize affine and dual affine quermassintegrals as averages on flag manifolds (where the Grassmannian can be considered as a special case). We establish affine and linear invariance properties and extend fundamental results to this new setting. In particular, we prove several affine isoperimetric inequalities from convex geometry and their approximate reverse forms. We also introduce functional forms of these quantities and establish corresponding inequalities.

The (φ,ψ) Orlicz mixed Petty bodies

In this paper, the homogeneous (φ,ψ) Orlicz mixed affine and geominimal surface areas are introduced. Their homogeneity, affine invariance and affine isoperimetric inequalities are established. Under some assumptions, the existence and uniqueness of the homogeneous (φ,ψ) Orlicz mixed Petty bodies are established. Base on these, the continuity of the homogeneous Orlicz mixed geominimal surface areas is given. Moreover, the nonhomogeneous (φ,ψ) Orlicz mixed affine and geominimal surface areas are defined, and the continuity of the nonhomogeneous (φ,ψ) Orlicz mixed geominimal surface for some special φ and ψ is obtained as well.

Invariant kinematic formulas and inequalities for chord power integrals in integral geometry

In this paper, we first introduce the history, developing process, recent progress, andworks contributed by Chinese researchers in integral and convex geometry.Then we show our recent works in invariant kinematicformulas and chord power integrals. The affine isoperimetric inequalities on the set of convex bodies, closely related to integral geometry, are also addressed.

Lp$L_p$‐cosine transform of the (p,q)${(p,q)}$‐th dual curvature measure

AbstractThe ‐th dual curvature measure was introduced by Lutwak, Yang and Zhang. In this paper, we study the cosine transform of the ‐th dual curvature measure which defines a new convex body. We prove that the new convex body unifies the Petty projection body and the centroid body. The corresponding affine isoperimetric inequality is also obtained. It is an extension of the known Petty projection inequality.

Symplectically convex and symplectically star-shaped curves: a variational problem

In this article, we propose a generalization of the 2-dimensional notions of convexity resp. being star-shaped to symplectic vector spaces. We call such curves symplectically convex resp. symplectically star-shaped. After presenting some basic results, we study a family of variational problems for symplectically convex and symplectically star-shaped curves which is motivated by the affine isoperimetric inequality. These variational problems can be reduced back to two dimensions. For a range of the family parameter, extremal points of the variational problem are rigid: they are multiply traversed conics. For all family parameters, we determine when non-trivial first- and second-order deformations of conics exist. In the last section, we present some conjectures and questions and two galleries created with the help of a Mathematica applet by Gil Bor.

Orlicz affine isoperimetric inequalities for star bodies

The original goal of this paper is to extend the affine isoperimetric inequality and Steiner type inequality of Orlicz projection bodies (which originated to Lutwak, Yang, and Zhang [32]), from convex bodies to Lipschitz star bodies (whose radial functions are locally Lipschitz).In order to achieve it, we investigate the graph functions of the given Lipschitz star body K: Along almost all directions u, we can define the graph functions on an open dense subset of the orthogonal projection of K onto u⊥.

Blaschke–Santaló inequality for many functions and geodesic barycenters of measures

Motivated by the geodesic barycenter problem from optimal transportation theory, we prove a natural generalization of the Blaschke–Santaló inequality and the affine isoperimetric inequalities for many sets and many functions. We derive from it an entropy bound for the total Kantorovich cost appearing in the barycenter problem. We also establish a “pointwise Prékopa–Leindler inequality” and show a monotonicity property of the multimarginal Blaschke–Santaó functional.

Sharp affine isoperimetric inequalities for the volume decomposition functionals of polytopes

The nth power of the volume functional Vn of polytopes P in Rn, according to dimensions of the spaces spanned by any n unit outer normal vectors of P, is decomposed into n homogeneous polynomials of degree n. A set of new sharp affine isoperimetric inequalities for these volume decomposition functionals in R3 are established, which essentially characterize the geometric and algebraic structures of polytopes.

General Affine Invariances Related to Mahler Volume

AbstractGeneral affine invariances related to Mahler volume are introduced. We establish their affine isoperimetric inequalities by using a symmetrization scheme that involves a total of $2n$ elaborately chosen Steiner symmetrizations at a time. The necessity of this scheme, as opposed to the usual Steiner symmetrization, is demonstrated with an example (see the Appendix).

On Complex L_p affine isoperimetric inequalities

On Complex L_p affine isoperimetric inequalities

On L_p intersection mean ellipsoids and affine isoperimetric inequalities

On L_p intersection mean ellipsoids and affine isoperimetric inequalities

Complex Lp affine isoperimetric inequalities

Sharp complex Lp affine isoperimetric inequalities are established for the entire class of complex Lp projection bodies and the entire class of complex Lp moment bodies.

Functional Geominimal Surface Area and Its Related 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.

Erratum

Erratum

Sharp affine isoperimetric inequalities for the Minkowski first mixed volume

Sharp affine isoperimetric inequalities for the Minkowski first mixed volume

Affine isoperimetric inequalities for intersection mean ellipsoids

A variational formula for Lutwak’s dual affine quermassintegrals $${\widetilde{\Lambda }}_{j}$$ of convex bodies in $${\mathbb {R}}^n$$ is established when $$1\le j\le n-1.$$ Using new ellipsoids associated with the intersection functions of convex bodies, we prove several sharp affine isoperimetric inequalities.

Complex affine isoperimetric inequalities

Complex extensions of the Petty projection inequality and the Busemann–Petty centroid inequality are established.

Orlicz Addition for Measures and an Optimization Problem for the -divergence

AbstractThis paper provides a functional analogue of the recently initiated dual Orlicz–Brunn–Minkowski theory for star bodies. We first propose the Orlicz addition of measures, and establish the dual functional Orlicz–Brunn–Minkowski inequality. Based on a family of linear Orlicz additions of two measures, we provide an interpretation for the famous $f$-divergence. Jensen’s inequality for integrals is also proved to be equivalent to the newly established dual functional Orlicz–Brunn–Minkowski inequality. An optimization problem for the $f$-divergence is proposed, and related functional affine isoperimetric inequalities are established.

A New Affine Invariant Geometric Functional for Polytopes and Its Associated Affine Isoperimetric Inequalities

Abstract A new affine invariant geometric functional for convex polytopes is introduced. Some new sharp affine isoperimetric inequalities are established for this new functional, which are extensions of Lutwak–Yang–Zhang’s results on their celebrated cone-volume functional.

Log-Minkowski inequalities for the L_{p}-mixed quermassintegrals

Böröczky et al. proposed the log-Minkowski problem and established the plane log-Minkowski inequality for origin-symmetric convex bodies. Recently, Stancu proved the log-Minkowski inequality for mixed volumes; Wang, Xu, and Zhou gave the L_{p} version of Stancu’s results. In this paper, we define the L_{p}-mixed quermassintegrals probability measure and obtain the log-Minkowski inequality for the L_{p}-mixed quermassintegrals. As its application, we establish the L_{p}-mixed affine isoperimetric inequality. In addition, we also consider the dual log-Minkowski inequalities for the L_{p}-dual mixed quermassintegrals.