Related papers: Square-densities, and volume forms
We use Papasoglu's method of area-minimizing separating sets to give an alternative proof, and explicit constants, for the following theorem of Guth and Braun--Sauer: If $M$ is a closed, oriented, $n$-dimensional manifold, with a Riemannian…
The classical result of Cauchy's surface area formula states that the surface area of the boundary $\partial K=\Sigma$ of any $n$-dimensional convex body in the $n$-dimensional Euclidean space $\mathbb{R}^n$ can be obtained by the average…
In this paper we introduce two new notions of sectional curvature for Riemannian manifolds with density. Under both notions of curvature we classify the constant curvature manifolds. We also prove generalizations of the theorems of…
It is proved that the volume of spherical or hyperbolic simplices, when considered as a function of the dihedral angles, can be extended continuously to degenerated simplices.
We show how to compute the circular area invariant of planar curves, and the spherical volume invariant of surfaces, in terms of line and surface integrals, respectively. We use the Divergence Theorem to express the area and volume…
We begin by studying the surface area of an ellipsoid in n-dimensional Euclidean space as the function of the lengths of the semi-axes. We write down an explicit formula as an integral over the unit sphere in n-dimensions and use this…
Given a closed oriented manifold or more generally a group homology class, we introduce the spherical Plateau problem, which is a variational problem corresponding to a topological invariant called the spherical volume. In principle, its…
We establish a lower bound for the surface area of a closed, convex hypersurface in Euclidean space in terms of its displacement under continuous maps. As a result, a hypothesized lower bound for the volume of a Riemannian $n$-sphere,…
We propose an approach to find constant curvature metrics on triangulated closed 3-manifolds using a finite dimensional variational method whose energy function is the volume. The concept of an angle structure on a tetrahedron and on a…
A general approach to compute the spherical measure of submanifolds in homogeneous groups is provided. We focus our attention on the homogeneous tangent space, that is a suitable weighted algebraic expansion of the submanifold. This space…
We derive several mean value formulae on manifolds, generalizing the classical one for harmonic functions on Euclidean spaces as well as later results of Schoen-Yau, Michael-Simon, etc, on curved Riemannian manifolds. For the heat equation…
We study the surface area of an ellipsoid in n-dimensional Euclidean space as the function of the lengths of their major semi-axes. We write down an explicit formula as an integral over the unit sphere, use the formula to derive convexity…
It is shown that for any ensemble, whether classical or quantum, continuous or discrete, there is only one measure of the "volume" of the ensemble that is compatible with several basic geometric postulates. This volume measure is thus a…
Given five points in a three-dimensional euclidean space, one can consider five tetrahedra, using those points as vertices. We present a pentagon-like formula containing the product of three volumes of those tetrahedra in its l.h.s. and the…
A ball polyhedron is a finite intersection of congruent balls in $\mathbb{R}^3$. These shapes arise in various contexts in discrete and convex geometry. We focus on Reuleaux polyhedra, the subclass of ball polyhedra whose centers and…
We consider the problem of determination of a volume of some bounded space-like hypersurfaces in the case of spherically symmetric spacetimes. In the case when the hypersurfaces is cut or bounded by a light-like hypersurface the problem may…
Given a Riemannian manifold M and a hypersurface H in M, it is well known that infinitesimal convexity on a neighborhood of a point in H implies local convexity. We show in this note that the same result holds in a semi-Riemannian manifold.…
We have derived an analytical formulation for estimating the volume of geometries enclosed by implicitly defined surfaces. The novelty of this work is due to two aspects. First we provide a general analytical formulation for all…
We derive an explicit formula for the volume of a regular simplex in the hyperbolic space of any dimension.
In shape analysis, the concept of shape spaces has always been vague, requiring a case-by-case approach for every new type of shape. In this paper, we give a general definition for an abstract space of shapes in a manifold. This notion…