Related papers: How to compute the volume in low dimension?
Magnitude is a numerical invariant of compact metric spaces, originally inspired by category theory and now known to be related to myriad other geometric quantities. Generalizing earlier results in $\ell_1^n$ and Euclidean space, we prove…
We give the sharp lower bound of the volume product of three dimensional convex bodies which are invariant under a discrete subgroup of $O(3)$ in several cases. We also characterize the convex bodies with the minimal volume product in each…
A fast method of an arbitrary high order for approximating volume potentials is proposed, which is effective also in high dimensional cases. Basis functions introduced in the theory of approximate approximations are used. Results of…
We analyse the maximum achievable rate of sustained computation for a given convex region of three dimensional space subject to geometric constraints on power delivery and heat dissipation. We find a universal upper bound across both…
We discuss how well a given convex body B in a real d-dimensional vector space V can be approximated by a set X for which the membership question: ``given an x in V, does x belong to X?'' can be answered efficiently (in time polynomial in…
We give a deterministic method of quasi-polynomial complexity to approximate the volume of the intersection of the unit hypercube with two specific sets. The method can actually be applied (without losing the quasi-polynomial complexity) to…
Enclosing depth is a recently introduced depth measure which gives a lower bound to many depth measures studied in the literature. So far, enclosing depth has only been studied from a combinatorial perspective. In this work, we give the…
Recently, motivated by the rapid increase of the data size in various applications, Monemizadeh [APPROX'23] and Driemel, Monemizadeh, Oh, Staals, and Woodruff [SoCG'25] studied geometric problems in the setting where the only access to the…
We study the expected volume of random polytopes generated by taking the convex hull of independent identically distributed points from a given distribution. We show that for log-concave distributions supported on convex bodies, we need at…
Let us consider the set of all joint probabilities generated by local binary measurements on two separated quantum systems of a given local dimension d. We address the question of whether the shape of this quantum body is convex or not. We…
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.
Consider the random polytope, that is given by the convex hull of a Poisson point process on a smooth convex body in $\mathbb{R}^d$. We prove central limit theorems for continuous motion invariant valuations including the Will's functional…
Simulating quantum systems in a finite volume is a powerful theoretical tool to extract information about them. Real-world properties of the system are encoded in how its discrete energy levels change with the size of the volume. This…
Computing the volume of a polytope in high dimensions is computationally challenging but has wide applications. Current state-of-the-art algorithms to compute such volumes rely on efficient sampling of a Gaussian distribution restricted to…
In a $d$-dimensional convex body $K$, for $n \leq d+1$, random points $X_0, \dots, X_{n-1}$ are chosen according to the uniform distribution in $K$. Their convex hull is a random $(n-1)$-simplex with probability $1$. We denote its…
Computing mixed volume of convex polytopes is an important problem in computational algebraic geometry. This paper establishes sufficient conditions under which the mixed volume of several convex polytopes exactly equals the normalized…
We derive an exact formula for the volume fraction of an inclusion in a body when the inclusion and the body are linearly elastic materials with the same shear modulus. Our formula depends on an appropriate measurement of the displacement…
The size estimation problem in electrical impedance tomography is considered when the conductivity is a complex number and the body is two-dimensional. Upper and lower bounds on the volume fraction of the unknown inclusion embedded in the…
The problem of determining the volume of a tubular neighbourhood has a long and rich history. Bounds on the volume of neighbourhoods of algebraic sets have turned out to play an important role in the probabilistic analysis of condition…
The volume density of a hyperbolic link is defined as the ratio of hyperbolic volume to crossing number. We study its properties and a closely-related invariant called the determinant density. It is known that the sets of volume densities…