Volume Inequalities for Sets Associated with Convex Bodies
نویسندگان
چکیده
This paper deals with inequalities for the volume of a convex body and the volume of the projection body, the L-centroid body, and their polars. Examples are the Blaschke-Santaló inequality, the Petty and Zhang projection inequalities, the Busemann-Petty inequality. Other inequalities of the same type are still at the stage of conjectures. The use of special continuous movements of convex bodies provides a general approach to this subject. A family of inequalities, depending on a parameter p ≥ 1 and proved by Lutwak for p = 1 and p = 2, is obtained.
منابع مشابه
On Volume Product Inequalities for Convex Sets
The volume of the polar body of a symmetric convex set K of Rd is investigated. It is shown that its reciprocal is a convex function of the time t along movements, in which every point of K moves with constant speed parallel to a fixed direction. This result is applied to find reverse forms of the Lp-Blaschke-Santaló inequality for two-dimensional convex sets.
متن کاملA probabilistic take on isoperimetric-type inequalities
We extend a theorem of Groemer’s on the expected volume of a random polytope in a convex body. The extension involves various ways of generating random convex sets. We also treat the case of absolutely continuous probability measures rather than convex bodies. As an application, we obtain a new proof of a recent result of Lutwak, Yang and Zhang on the volume of Orlicz-centroid bodies.
متن کاملVolume difference inequalities for the projection and intersection bodies
In this paper, we introduce a new concept of volumes difference function of the projection and intersection bodies. Following this, we establish the Minkowski and Brunn-Minkowski inequalities for volumes difference function of the projection and intersection bodies.
متن کاملThin Partitions: Isoperimetric Inequalities and Sampling Algorithms for some Nonconvex Families
Star-shaped bodies are an important nonconvex generalization of convex bodies (e.g., linear programming with violations). Here we present an efficient algorithm for sampling a given star-shaped body. The complexity of the algorithm grows polynomially in the dimension and inverse polynomially in the fraction of the volume taken up by the kernel of the star-shaped body. The analysis is based on a...
متن کاملVolume Inequalities and Additive Maps of Convex Bodies
Analogs of the classical inequalities from the Brunn Minkowski Theory for rotation intertwining additive maps of convex bodies are developed. We also prove analogs of inequalities from the dual Brunn Minkowski Theory for intertwining additive maps of star bodies. These inequalities provide generalizations of results for projection and intersection bodies. As a corollary we obtain a new Brunn Mi...
متن کامل