Related papers: Central diagonal sections of the $n$-cube

In his paper "On the Schlafli differential equality", J. Milnor conjectured that the volume of n-dimensional hyperbolic and spherical simplices, as a function of the dihedral angles, extends continuously to the closure of the space of…

This paper deals with the union set of a stationary Poisson process of cylinders in $\mathbb{R}^n$ having an $(n-m)$-dimensional base and an $m$-dimensional direction space, where $m\in\{0,1,\ldots,n-1\}$ and $n\geq 2$. The concept…

This note shows that an interesting property arises when considering the relation between the hypersphere volumes at dimensions $n+1$ and $n$, if the hyperspheres circumscribe unitary hypercubes in $n+1$ and $n$ dimensions, respectively .…

We determine the minimal volume of arithmetic hyperbolic orientable n-dimensional orbifolds (compact and non-compact) for every odd dimension n>3. Combined with the previously known results it solves the minimal volume problem for…

The volume function defined by a domain in Euclidean space $\mathbb{R}^n$ is the function on the space of affine hyperplanes equal to volumes cut by these hyperplanes from the domain. The study of these functions originates from the works…

The intrinsic volumes are measures of the content of a convex body. This paper uses probabilistic and information-theoretic methods to study the sequence of intrinsic volumes of a convex body. The main result states that the intrinsic…

We give the first nontrivial upper and lower bounds on the maximum volume of an empty axis-parallel box inside an axis-parallel unit hypercube in $\RR^d$ containing $n$ points. For a fixed $d$, we show that the maximum volume is of the…

We calculate the one-loop quantum corrections to the mass and central charge of N=2 and N=4 supersymmetric monopoles in 3+1 dimensions. The corrections to the N=2 central charge are finite and due to an anomaly in the conformal central…

We show that the rate of convergence on the approximation of volumes of a convex symmetric polytope P in R^n by its dual L_{p$-centroid bodies is independent of the geometry of P. In particular we show that if P has volume 1,…

We prove a randomized version of the generalized Urysohn inequality relating mean-width to the other intrinsic volumes. To do this, we introduce a stochastic approximation procedure that sees each convex body K as the limit of intersections…

For a regular sub-Riemannian manifold we study the Radon-Nikodym derivative of the spherical Hausdorff measure with respect to a smooth volume. We prove that this is the volume of the unit ball in the nilpotent approximation and it is…

The aim of this work is to expose some asymptotic series associated to some expressions involving the volume of the n-dimensional unit ball. All proofs and the methods used for improving the classical inequalities announced in the final…

We present a deterministic polynomial-time algorithm for estimating the volume of a hypercube intersected by a fixed number of constraints of the type $f(x) \leq b$, where $f$ is the sum of univariate functions that are each nonnegative,…

We determine the maximal non-central hyperplane sections of the n-dimensional $l_1$-ball if the fixed distance of the hyperplane to the origin is between $1 / \sqrt 3$ and $1 / \sqrt 2$. This adds to a result of Liu and Tkocz who considered…

We establish a monotonicity-type property of volume of central hyperplane sections of the 1-symmetric convex bodies, with applications to chessboard cutting. We parallel this for projections with a new convexity-type property for Rademacher…

We show the asymptotics of the volume density function in the class of central harmonic manifolds can be specified arbitrarily and do not determine the geometry.

Let $\Gamma$ be a nonuniform lattice acting on real hyperbolic n-space. We show that in dimension greater than or equal to 4, the volume of a representation is constant on each connected component of the representation variety of $\Gamma$…

Let $M$ be an open (i.e. complete and noncompact) manifold with nonnegative Ricci curvature. In this paper, we study whether the volume growth order of $M$ is always greater than or equal to the dimension of some (or every) asymptotic cone…

The orthogonal projection of a 4-cube onto a uniform random 3-subspace in R^4 is a convex 3-polyhedron P with 14 vertices almost surely. Three numerical characteristics of P -- volume, surface area and mean width -- are studied. These…

We study monotonicity and 1-dimensional symmetry for positive solutions with algebraic growth of the following elliptic system: \[ \begin{cases} -\Delta u = -u v^2 & \text{in $\R^N$}\\ -\Delta v= -u^2 v & \text{in $\R^N$}, \end{cases} \]…