相关论文: A new algorithm for the volume of a convex polytop…

We prove the theorem mentioned in the title, for ${\mathbb{R}}^n$, where $n \ge 3$. The case of the simplex was known previously. Also, the case $n=2$ was settled, but there the infimum was some well-defined function of the side lengths. We…

We give efficient algorithms for volume sampling, i.e., for picking $k$-subsets of the rows of any given matrix with probabilities proportional to the squared volumes of the simplices defined by them and the origin (or the squared volumes…

In this work we bring together tools and ideology from two different fields, Symplectic Geometry and Asymptotic Geometric Analysis, to arrive at some new results. Our main result is a dimension-independent bound for the symplectic capacity…

We propose an exact polynomial algorithm for a resource allocation problem with convex costs and constraints on partial sums of resource consumptions, in the presence of either continuous or integer variables. No assumption of strict…

We consider the {\em Shaped Partition Problem} of partitioning $n$ given vectors in real $k$-space into $p$ parts so as to maximize an arbitrary objective function which is convex on the sum of vectors in each part, subject to arbitrary…

We present a new algorithm to calculate exact hypervolumes. Given a set of $d$-dimensional points, it computes the hypervolume of the dominated space. Determining this value is an important subroutine of Multiobjective Evolutionary…

The free sum is a basic geometric operation among convex polytopes. This note focuses on the relationship between the normalized volume of the free sum and that of the summands. In particular, we show that the normalized volume of the free…

The approximability of a convex body is a number which measures the difficulty to approximate that body by polytopes. We prove that twice the approximability is equal to the volume entropy for a Hilbert geometry in dimension two end three…

The asymptotic volume of the polytope of symmetric stochastic matrices can be determined by asymptotic enumeration techniques as in the case of the Birkhoff polytope. These methods can be extended to polytopes of symmetric stochastic…

Let n >= 2 be an integer and consider the set T_n of n by n permutation matrices pi for which pi_{ij}=0 for j>=i+2. In this paper we study the convex hull of T_n, which we denote by P_n. P_n is a polytope of dimension binom{n}{2}. Our main…

We construct high order symmetric volume-preserving methods for the relativistic dynamics of a charged particle by the splitting technique with processing. Via expanding the phase space to include time $t$, we give a more general…

In this thesis, a new approach for constructing subdivision algorithms for generalized quadratic and cubic B-spline subdivision for subdivision surfaces and volumes is presented. First, a catalog of quality criteria for these subdivision…

We present new differentially private algorithms for learning a large-margin halfspace. In contrast to previous algorithms, which are based on either differentially private simulations of the statistical query model or on private convex…

We compute the intrinsic volumes of the cone of positive semidefinite matrices over the real numbers, over the complex numbers, and over the quaternions, in terms of integrals related to Mehta's integral. Several applications for the…

This article introduces a representation of dynamic meshes, adapted to some numerical simulations that require controlling the volume of objects with free boundaries, such as incompressible fluid simulation, some astrophysical simulations…

These lectures on the combinatorics and geometry of 0/1-polytopes are meant as an \emph{introduction} and \emph{invitation}. Rather than heading for an extensive survey on 0/1-polytopes I present some interesting aspects of these objects;…

The mixed volume counts the roots of generic sparse polynomial systems. Mixed cells are used to provide starting systems for homotopy algorithms that can find all those roots, and track no unnecessary path. Up to now, algorithms for that…

In this note, we will consider two classical volume problems related to elliptic integrals. The first problem has a neat formula by means of elliptic integrals. We remade it with details. In the second problem, we found a messy formula. On…

While recent work suggests that quantum computers can speed up the solution of semidefinite programs, little is known about the quantum complexity of more general convex optimization. We present a quantum algorithm that can optimize a…

For a convex lattice polytope $P\subset \mathbb R^d$ of dimension $d$ with vertices in $\mathbb Z^d$, denote by $L(P)$ its discrete volume which is defined as the number of integer points inside $P$. The classical result due to Ehrhart says…