Related papers: The volume enclosed by an n-dimensional Lam\'e cur…
The manuscript provides formulas for the volume of a body defined by the intersection of a solid cone and a solid sphere as a function of the sphere radius, of the distance between cone apex and sphere center, and of the cone aperture…
We derive bounds on the volume of an inclusion in a body in two or three dimensions when the conductivities of the inclusion and the surrounding body are complex and assumed to be known. The bounds are derived in terms of average values of…
General results on convex bodies are reviewed and used to derive an exact closed-form parametric formula for the Minkowski sum boundary of $m$ arbitrary ellipsoids in $N$-dimensional Euclidean space. Expressions for the principal curvatures…
This paper deals with the question of how to calculate the volume of a body in the three-dimensional Euclidean space when it is cut into slices perpendicular to a given curve. The answer is provided by a formula that can be considered as a…
The volume of the unit ball of the Lebesgue sequence space $\ell_p^m$ is very well known since the times of Dirichlet. We calculate the volume of the unit ball in the mixed norm $\ell^n_q(\ell_p^m)$, whose special cases are nowadays popular…
We compute explicitly the Riemannian volume, with respect to the Fubini-Study metric, of a domain bounded by a Hermitian quadric in complex projective space. The volume is a rational function of the eigenvalues of the defining quadratic…
We provide two algorithms for computing the volume of a convex polytope with half-space representation {x>=0; Ax <=b} for some (m,n) matrix A and some m-vector b. Both algorithms have a O(n^m) computational complexity which makes them…
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…
If the n-dimensional unit sphere is covered by finitely many spherically convex bodies, then the sum of the inradii of these bodies is at least {\pi}. This bound is sharp, and the equality case is characterized.
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…
In this paper we deal with problems concerning the volume of the convex hull of two "connecting" bodies. After a historical background we collect some results, methods and open problems, respectively.
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…
We consider the problem of estimating the distance between two bodies of volume $\varepsilon$ located inside a $n$-dimensional ball $U$ of unit volume for $n\to\infty$. Let $A$ be a closed set with a smooth boundary of the volume…
Let $K$ be a convex body in $\mathbb{R}^d$ which slides freely in a ball. Let $K^{(n)}$ denote the intersection of $n$ closed half-spaces containing $K$ whose bounding hyperplanes are independent and identically distributed according to a…
Estimating the volume of a convex body is a canonical problem in theoretical computer science. Its study has led to major advances in randomized algorithms, Markov chain theory, and computational geometry. In particular, determining the…
In this note we examine the volume of the convex hull of two congruent copies of a convex body in Euclidean $n$-space, under some subsets of the isometry group of the space. We prove inequalities for this volume if the two bodies are…
For every large enough $n$, we explicitly construct a body of constant width $2$ that has volume less than $0.9^n \text{Vol}(\mathbb{B}^{n}$), where $\mathbb{B}^{n}$ is the unit ball in $\mathbb{R}^{n}$. This answers a question of…
The problem of bounding of the distance between the two bodies of volume $\varepsilon$ located inside the $n$-dimensional body $B$ of unit volume where $n \to \infty$ is considered. In some cases such distances are bounded by function…
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…
Numerical tests of volume formulae are presented to compute efficiently the volume enclosed between flux surfaces for integrable 3D vector fields with various degrees of symmetry. In the process, a new case is proposed and tested.