# Approximating The Volume Of Convex Bodies

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### On the Combinatorial Complexity of Approximating - DROPS

by S Arya 2016 Cited by 17 Approximating convex bodies succinctly by convex polytopes is a lattice polytope2 is O(V (d−1)/(d+1)), where V is the volume of the polytope [2, 8]. This.

### A Concentration Inequality for Random Polytopes, Dirichlet

by S Hoehner 2017 Cited by 2 Random polytopes and approximation and convex bodies and sphere the volume of the symmetric difference of a given convex body K in Rn

### Stochastical approximation of smooth convex bodies

by M Reitzner 2004 Cited by 43 Abstract. A random polytope is the convex hull of n random points in the interior of a convex body K. The expectation of the fth intrinsic volume of a random

### Random Approximation of Convex Bodies - Institut für

by S Kunis Cited by 1 Random Approximation of Convex Bodies: Monotonicity of the Volumes of Random Tetrahedra. Stefan Kunis, Benjamin Reichenwallner and Matthias Reitzner.

### Approximating the volume of a convex body - MIT

29 Oct 2009 Exactly computing the volume a convex body is known to be #P-hard, so the fact that we can approximate its volume in P is surprising the kind

### Algorithms for volume approximation of convex bodies

by IZ Emiris 2013 Cited by 3 algorithms for approximating the volume of polytopes given as an volume of a convex body K ⊂ Rd. Their ϵ-approximation algorithm with

### Volume approximation of convex bodies by polytopesâ ﬂa

by Y GORDON 1990 contained in (or containing) a given convex body K in R , so that the ratio of the volumes. K P/K! (or P KV K) is smaller than f(d)/n2/(1-1),. 1. Introduction.

### CONVEX GEOMETRY These are the lecture notes of a course

by A COLESANTI Every convex body can be approximated by a sequence of convex polytopes. In other following question: how to compute the volume of a convex body?

### Heat Flow and a Faster Algorithm to Compute the Surface

by M Belkin Cited by 13 ple algorithm for approximating the surface area of a convex body given by a membership oracle. Our method has a complexity of O∗(n4), where n is the

### Approximation of convex bodies by polytopes Viktor V´ıgh

by V Vígh 2010 polytopes, the other is approximation of convex bodies by random poly- topes. The dissertation I. Bárány, F. Fodor, V. V´ıgh: Intrinsic volumes of inscribed ran-.

### On the approximation of a convex body by its radial mean

In this paper, we consider the approximation problem on the volume of a convex body K in Rn by those of its radial mean bodies RpK. Specifically, we establish

### Summary - NUS Computing

approximating the volume of a convex body Input to the algorithm is an oracle (which decides the membership of a point in a fixed convex body), two spheres

### Best and random approximation of convex bodies by polytopes

How well can a convex body be approximated by a polytope? µ is the surface area measure on ∂K The expected volume of such a random polytope is.

### Approximation of General Smooth Convex Bodies

by K Böröczky Jr 2000 Cited by 73 approximating o-symmetric polytope with 2n vertices (2n facets) then (see. [15]). $BM (K convex body whose affine surface area is positive. M. Ludwig has

### A near-optimal algorithm for approximating the John Ellipsoid

by MB Cohen 2019 Cited by 5 if we want to get (1 − ϵ)-approximation to the maximal volume, we shall set η Several Markov chains for sampling convex bodies have well understood

### A random polynomial time algorithm for approximating the

by MDAFR Kannan Cited by 881 We present a randomised polynomial time algorithm for approximating the volume of a convex body K in n-dimensional Euclidean space. The.

### MAHLER'S CONJECTURE IN CONVEX - SMARTech

by P Hupp 2010 As we later want to approximate the conjectured minimizers of the Mahler volume, we will introduce metrics for the family of convex bodies. This allows to judge the

### Random Polytopes, Convex Bodies, and Approximation

by I Bárány Cited by 72 Assume K ⊂ Rd is a convex body and Xn ⊂ K is a random sample of n uniform number of k-dimensional faces, fk(Kn), of Kn, or the volume missed by Kn,.

### Polytopes with Vertices Chosen Randomly from the Boundary

by C Schütt Cited by 76 How well can a convex body be approximated by a polytope? This is a central interpretation of the p-affine surface area in terms of random polytopes. It was a

### A RANDOM POLYNOMIAL TIME ALGORITHM FOR

Abstract. We present a randomised polynomial time algorithm for approximating the volume of a convex body t in n-dimensional Euclidean space. The proof of

### A random polynomial-time algorithm for approximating the

by M DYER 1991 Cited by 881 Abstract. A randomized polynomial-time algorithm for approximating the volume of a convex body K in n-dimensional Euclidean spaceis presented. The proof of

### Approximating the volume of convex bodies

by U Betke 1993 Cited by 26 Abstract. It is a well-known fact that for every polynomial-time algorithm which gives an upper bound 17(K) and a lower bound V(K) for the volume of a convex set.

### Estimating the Sizes of Convex Bodies from Projections - Wiley

by U Betke 1983 Cited by 46 r-volume of the image of the at least r-dimensional compact convex set K Now stereometry is concerned with problems of estimating the size of a body (in.

### Asymptotic estimates for best and stepwise approximation of

by PM Gruber 1993 Cited by 90 of convex bodies II. Peter M. Gruber. (Communicated by Karl Strambach). Abstract. We derive an asymptotic formula for the difference of the volumes of a smooth

### APPROXIMATION OF CONVEX BODIES BY TRIANGLES X

by M Lassak 1992 Cited by 21 Volume 115, Number 1, May 1992 We show that for every plane convex body C there exist a triangle Let C be a convex body in the Euclidean plane E2

### Efficient Random-Walk Methods for Approximating - HAL-Inria

by IZ Emiris 2018 Cited by 33 a convex body in general dimension or, more particularly, of a polytope. poly-time algorithm in [15] approximates the volume of a convex.

### Approximating the Centroid is Hard - UC Davis Math

by L Rademacher 2007 Cited by 54 troid of a convex body cuts it into two parts such that each has a volume that is at least a 1/e fraction of the volume of the body. Permission to make digital or hard

### Asymptotic approximation of smooth convex bodies by

by S Glasauer 1996 Cited by 25 Best approximation, smooth convex body, inscribed polytope, Hausdorff of K. In the following, the constants κk and ϑk are, respectively, the volume of the.

### Approximating the surface volume of convex bodies

by H Narayanan Grötschel, Lovász and Schrijver (1987) mention computing the surface volume of a convex body to be an open problem. The first and to our knowledge only

### Fine approximation of convex bodies by polytopes - Johns

by M Naszódi 2020 Cited by 4 Fine approximation of convex bodies by polytopes. Márton Naszódi, Fedor Nazarov, Dmitry Ryabogin. American Journal of Mathematics, Volume 142, Number 3

### Thrifty approximations of convex bodies by polytopes

by A Barvinok 2012 Cited by 41 Key words and phrases. approximation, convex body, polytope, Chebyshev the John decomposition of the identity operator and the minimum volume ellipsoid.

### Estimating the volume of a convex body. - Imaginary

by N Baldin Cited by 2 We explain how statistics can be used not only to approximate the volume of the convex body, but also its shape. 1 Calculating the volume in analytic geometry.

### Approximating the volume of unions and intersections of high

by K Bringmann Cited by 175 Our algorithm also allows to approximate efficiently the volume of the union of convex bodies given by weak membership oracles. For the analogous problem of

### A Fast and Practical Method to Estimate Volumes of Convex

by CJ Ge Cited by 21 cult to calculate the exact volume. But in many cases, it suffices to have an approximate value. Volume estimation methods for convex bodies have been

### Intrinsic and Dual Volume Deviations of Convex Bodies and

by F Besau Cited by 5 We establish estimates for the asymptotic best approximation of the Euclidean unit ball by polytopes under a notion of distance induced by the intrinsic volumes.

### Oracle-polynomial-time approximation of largest - mediaTUM

by A Briedena 2000 Cited by 7 A convex body (or simply body) in En is an n-dimensional compact oracle-polynomial-time algorithms for approximating the volumes of largest simplices in.

### Estimating the Volume of the Solution Space of SMT(LIA) - MDPI

by W Gao 2018 Cited by 2 In general, a SMT(LIA) formula may imply m convex bodies, and the maximal value of m is: 2m − 1 where m is the number of Boolean variables.

### Fine approximation of convex bodies by polytopes

by M NASZODI Cited by 4 arbitrary convex body K can be reduced to that of approximating a certain uniformly immediately from the general properties of the volume with respect to.

### A random polynomial time algorithm for approximating the

by M Dyer 1989 Cited by 881 mating the volume of a convex body in Euclidean such an oracle it is not possible to approximate the volume of a convex set within even a polynomial fac-.

### Asymptotic approximation of smooth convex bodies by

by M Ludwig Cited by 58 imply that among all convex bodies of given volume ellipsoids are asymptotically worst approximated by polytopes. As a second notion of distance we use the

### Approximating the volume of unions and - MPI-INF

by K Bringmann 2010 Cited by 175 Our algorithm also allows to efficiently approximate the volume of the union of convex bodies given by weak membership oracles. For the

### Learning convex bodies is hard - Association for

by N Goyal Cited by 15 approximate the volume of a convex body within a constant factor ([3], and see Section 5 here for a discussion). Note that known approximation algorithms for

### On the Combinatorial Complexity of Approximating Polytopes

by S Arya 2016 Cited by 17 Approximating convex bodies succinctly by convex polytopes is a fundamental problem in where V is the volume of the polytope [2,9].

### RANDOM WALKS IN A CONVEX BODY AND AN IMPROVED

by L Lovász Cited by 442 Abstract. We give a randomized algorithm using O(n7 log2 n) separation calls to approximate the volume of a convex body with a fixed relative error. The bound

### Approximation of Smooth Convex Bodies by Random - JSTOR

by K Böröczky Jr 2004 Cited by 31 The intersection of the supporting halfspaces at these random points is a random convex polyhedron. The expectations of its volume, its surface area and its mean

### How to compute the volume in high dimension?

by M Simonovits Cited by 74 Dyer, Frieze and Kannan gave a randomized polynomial approximation algorithm for the volume of a convex body K ⊆ Rn, given by a.

### How to estimate the volume of convex bodies

17 Jul 2010 How to estimate the volume of convex bodies Our aim here is to find a good approximation for the volume Vol(K) of K. Ideally, given ε > 0, we

### Volume Approximation and Sampling for Convex - arXiv

by A Chalkis 2020 Cited by 4 Abstract. Sampling from high dimensional distributions and volume approximation of convex bodies are fundamental operations that appear in

### THE MINIMAL VOLUME OF SIMPLICES CONTAINING A

by DE Galicer 2019 Cited by 2 Approximating a geometric body by a much simpler one results a very Volume ratio, Simplices, Convex Bodies, Isotropic Position, Random Simplices.

### Approximation of General Smooth Convex Bodies - CORE

by K Boroczky Jr 2000 Cited by 73 approximating o-symmetric polytope with 2n vertices (2n facets) then (see. [15]). $BM (K convex body whose affine surface area is positive. M. Ludwig has