An affine Orlicz Polya-Szego principle

The author established the affine Orlicz Polya-Szego principle for log-concave functions and conjectured that the principle can be extended to the general Orlicz Sobolev functions. In this paper, we confirm this conjecture completely. An affine Orlicz Polya-Szego principle, which includes all the previous affine Polya-Szego principles as special cases, is formulated and proved. As a consequence, an Orlicz-Petty projection inequality for star bodies is established

Integral Geometry and its Applications

In recent years there has been a series of striking developments in modern integral geometry which has, in particular, lead to the discovery of new relations to several branches of pure and applied mathematics. A number of examples were presented at this meeting, e.g. the work of Bernig, Solanes, and Fu on kinematic formulas on complex projective and complex hyperbolic spaces, that of Schneider and Vedel Jensen on tensor valuations and a series of results on convex body valued valuations by Abardia, Ludwig, Parapatits, and Wannerer

Sharp geometric inequalities for the general pp-affine capacity

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

The Petty projection inequality for sets of finite perimeter

The Petty projection inequality for sets of finite perimeter is proved. Our approach is based on Steiner symmetrization. Neither the affine Sobolev inequality nor the functional Minkowski problem is used in our proof. Moreover, for sets of finite perimeter, we prove the Petty projection inequality with respect to Steiner symmetrization

Minkowski valuations on lattice polytopes

A complete classification is established of Minkowski valuations on lattice polytopes that intertwine the special linear group over the integers and are translation invariant. In the contravariant case, the only such valuations are multiples of projection bodies. In the equivariant case, the only such valuations are generalized difference bodies combined with multiples of the newly defined discrete Steiner point.Comment: accepted in JEMS, 201

The floating body in real space forms

We carry out a systematic investigation on floating bodies in real space forms. A new unifying approach not only allows us to treat the important classical case of Euclidean space as well as the recent extension to the Euclidean unit sphere, but also the new extension of floating bodies to hyperbolic space. Our main result establishes a relation between the derivative of the volume of the floating body and a certain surface area measure, which we called the floating area. In the Euclidean setting the floating area coincides with the well known affine surface area, a powerful tool in the affine geometry of convex bodies

The Logarithmic Minkowski conjecture and the LpL_p-Minkowski Problem

The current state of art concerning the $L_p$ Minkowski problem as a Monge-Ampere equation on the sphere and Lutwak's Logarithmic Minkowski conjecture about the uniqueness of even solution in the $p=0$ case are surveyed and connections to many related problems are discussed

Factoring Sobolev inequalities through classes of functions

We recall two approaches to recent improvements of the classical Sobolev inequality. The first one follows the point of view of Real Analysis, while the second one relies on tools from Convex Geometry. In this paper we prove a (sharp) connection between them

Some Affine Invariants Revisited

We present several sharp inequalities for the SL(n) invariant $\Omega_{2,n}(K)$ introduced in our earlier work on centro-affine invariants for smooth convex bodies containing the origin. A connection arose with the Paouris-Werner invariant $\Omega_K$ defined for convex bodies $K$ whose centroid is at the origin. We offer two alternative definitions for $\Omega_K$ when $K \in C^2_+$. The technique employed prompts us to conjecture that any SL(n) invariant of convex bodies with continuous and positive centro-affine curvature function can be obtained as a limit of normalized $p$-affine surface areas of the convex body.Comment: 15 page

Convex Geometry and its Applications

The past 30 years have not only seen substantial progress and lively activity in various areas within convex geometry, e.g., in asymptotic geometric analysis, valuation theory, the $L_p$-Brunn-Minkowski theory and stochastic geometry, but also an increasing amount and variety of applications of convex geometry to other branches of mathematics (and beyond), e.g. to PDEs, statistics, discrete geometry, optimization, or geometric algorithms in computer science. Thus convex geometry is a flourishing and attractive field, which is also reflected by the considerable number of talented young mathematicians at this meeting