相关论文: General Measure Extensions of Projection Bodies

We prove sharp inequalities for the average number of affine diameters through the points of a convex body $K$ in ${\mathbb R}^n$. These inequalities hold if $K$ is either a polytope or of dimension two. An example shows that the proof…

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…

We give a refined Young inequality which generalizes the inequality by Zou--Jiang. We also show the upper bound for the logarithmic mean by the use of the weighted geometric mean and the weighted arithmetic mean. Furthermore, we show some…

In this paper, we develop a refined analysis of hypergeometric functions to establish sharp quantitative integral inequalities for a general family of conformally invariant extension operators and their adjoints. Our results extend the…

Using gnomonic projection and Poincar\'e model, we first define the spherical projection body and hyperbolic projection body in spherical space $\mathbb{S}^n$ and hyperbolic space $\mathbb{H}^n$, then define the spherical Steiner…

We establish new geometric inequalities comparing the volumes of sections and projections of a convex body, whose barycenter or Santal\'o point is at the origin, with those of its inner and outer regularizations. We also provide functional…

We consider two well-known problems: upper bounding the volume of lower dimensional ellipsoids contained in convex bodies given their John ellipsoid, and lower bounding the volume of ellipsoids containing projections of convex bodies given…

In this note prove the following Berwald-type inequality, showing that for any integrable log-concave function $f:\mathbb R^n\rightarrow[0,\infty)$ and any concave function $h:L\rightarrow\mathbb [0,\infty)$, where $L$ is the epigraph of…

The general volume of a star body, a notion that includes the usual volume, the $q$th dual volumes, and many previous types of dual mixed volumes, is introduced. A corresponding new general dual Orlicz curvature measure is defined that…

Metric projection operators can be defined in similar wayin Hilbert and Banach spaces. At the same time, they differ signifitiantly in their properties. Metric projection operator in Hilbert space is a monotone and nonexpansive operator. It…

For $n$ positive numbers ($a_k$, $1\leq k \leq n$), enhanced inequalities about the arithmetic mean ($A_n \equiv \frac{\sum_ka_k}{n}$) and the geometric mean ($G_n\equiv \sqrt[n]{\Pi_ka_k}$) are found if some numbers are known, namely,…

This article is a survey of recent results on slicing inequalities for convex bodies. The focus is on the setting of arbitrary measures in place of volume.

A comparison problem for volumes of convex bodies asks whether inequalities $f_K(\xi)\le f_L(\xi)$ for all $\xi\in S^{n-1}$ imply that $\vol_n(K)\le \vol_n(L),$ where $K,L$ are convex bodies in $\R^n,$ and $f_K$ is a certain geometric…

It is shown that every even, zonal measure on the Euclidean unit sphere gives rise to an isoperimetric inequality for sets of finite perimeter which directly implies the classical Euclidean isoperimetric inequality. The strongest member of…

Minkowski's Theorem asserts that every centered measure on the sphere which is not concentrated on a great subsphere is the surface area measure of some convex body, and, moreover, the surface area measure determines a convex body uniquely.…

This paper deals with weighted isoperimetric inequalities relative to cones of $\mathbb{R}^{N}$. We study the structure of measures that admit as isoperimetric sets the intersection of a cone with balls centered at the vertex of the cone.…

It is well known that isoperimetric inequalities imply in a very general measure-metric-space setting appropriate concentration inequalities. The former bound the boundary measure of sets as a function of their measure, whereas the latter…

Hayashi's Pinching Inequality, which establishes a matrix inequality between a semidefinite matrix and a multiple of its "pinched" version via a projective measurement, has found many applications in quantum information theory and beyond.…

Let $\nu$ be a self similar measure on $\mathbb{R}^d$, $d\geq 2$, and let $\pi$ be an orthogonal projection onto a $k$-dimensional subspace. We formulate a criterion on the action of the group generated by the orthogonal parts of the IFS on…

Analogues of the classical inequalities from the Brunn-Minkowski theory for rotation intertwining additive maps of convex bodies are developed. Analogues are also proved of inequalities from the dual Brunn-Minkowski theory for intertwining…