Topic

# Convex body

About: Convex body is a research topic. Over the lifetime, 3687 publications have been published within this topic receiving 64417 citations.

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1,492 citations

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01 Jan 2014

TL;DR: In this paper, an extension of Lagrange's multiplier rule to the case where the subsidiary conditions are inequalities instead of equations is considered, where only extrema of differentiable functions of a finite number of variables will be considered.

Abstract: This paper deals with an extension of Lagrange’s multiplier rule to the case, where the subsidiary conditions are inequalities instead of equations Only extrema of differentiable functions of a finite number of variables will be considered There may however be an infinite number of inequalities prescribed Lagrange’s rule for the situation considered here differs from the ordinary one, in that the multipliers may always be assumed to be positive This makes it possible to obtain sufficient conditions for the occurence or a minimum in terms of the first derivatives only

1,063 citations

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TL;DR: An algorithm for solving Integer Programming problems whose running time depends on the number n of variables as nOn by reducing an n variable problem to 2n5i/2 problems in n-i variables for some i greater than zero chosen by the algorithm.

Abstract: The paper presents an algorithm for solving Integer Programming problems whose running time depends on the number n of variables as nOn. This is done by reducing an n variable problem to 2n5i/2 problems in n-i variables for some i greater than zero chosen by the algorithm. The factor of On5/2 “per variable” improves the best previously known factor which is exponential in n. Minkowski's Convex Body theorem and other results from Geometry of Numbers play a crucial role in the algorithm. Several related algorithms for lattice problems are presented. The complexity of these problems with respect to polynomial-time reducibilities is studied.

841 citations

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TL;DR: The proof of correctness of the algorithm relies on recent theory of rapidly mixing Markov chains and isoperimetric inequalities to show that a certain random walk can be used to sample nearly uniformly from within K within Euclidean space.

Abstract: A randomized polynomial-time algorithm for approximating the volume of a convex body K in n-dimensional Euclidean space is presented. The proof of correctness of the algorithm relies on recent theory of rapidly mixing Markov chains and isoperimetric inequalities to show that a certain random walk can be used to sample nearly uniformly from within K.

702 citations

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TL;DR: This lemma is a general “Localization Lemma” that reduces integral inequalities over then-dimensional space to integral inequalities in a single variable and is illustrated by showing how a number of well-known results can be proved using it.

Abstract: We study the smallest number ?(K) such that a given convex bodyK in ?n can be cut into two partsK1 andK2 by a surface with an (n?1)-dimensional measure ?(K) vol(K1)·vol(K2)/vol(K). LetM1(K) be the average distance of a point ofK from its center of gravity. We prove for the "isoperimetric coefficient" that % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfKttLearuqr1ngBPrgarmWu51MyVXgatC% vAUfeBSjuyZL2yd9gzLbvyNv2CaeHbd9wDYLwzYbItLDharyavP1wz% ZbItLDhis9wBH5garqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbb% L8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpe% pae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaam% aaeaqbaaGcbaqegWuDJLgzHbYqV52CVXwzaGGbciaa-H8acqGGOaak% cqWGlbWscqGGPaqkcqGHLjYSdaWcaaqaaiGbcYgaSjabc6gaUjabik% daYaqaaiabd2eannaaBaaaleaacqaIXaqmaeqaaOGaeiikaGIaem4s% aSKaeiykaKcaaaaa!4EFC! $$\psi (K) \geqslant \frac{{\ln 2}}{{M_1 (K)}}$$ , and give other upper and lower bounds. We conjecture that our upper bound is the exact value up to a constant.
Our main tool is a general "Localization Lemma" that reduces integral inequalities over then-dimensional space to integral inequalities in a single variable. This lemma was first proved by two of the authors in an earlier paper, but here we give various extensions and variants that make its application smoother. We illustrate the usefulness of the lemma by showing how a number of well-known results can be proved using it.

489 citations