Related papers: Small volume bodies of constant width

Let $K$ be an $n$-dimensional symmetric convex body with $n \ge 4$ and let $K\dual$ be its polar body. We present an elementary proof of the fact that $$(\Vol K)(\Vol K\dual)\ge \frac{b_n^2}{(\log_2 n)^n},$$ where $b_n$ is the volume of the…

We conjecture that for every dimension n not equal 3 there exists a noncompact hyperbolic n-manifold whose volume is smaller than the volume of any compact hyperbolic n-manifold. For dimensions n at most 4 and n=6 this conjecture follows…

We study the volume ratio between projections of two convex bodies. Given a high-dimensional convex body $K$ we show that there is another convex body $L$ such that the volume ratio between any two projections of fixed rank of the bodies…

We prove several inequalities estimating the distance between volumes of two bodies in terms of the maximal or minimal difference between areas of sections or projections of these bodies. We also provide extensions in which volume is…

We consider a generalization of the hyperplane problem to arbitrary measures in place of volume and to sections of lower dimensions. We prove this generalization for unconditional convex bodies and for duals of bodies with bounded volume…

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…

A subset of the d-dimensional Euclidean space having nonempty interior is called a spindle convex body if it is the intersection of (finitely or infinitely many) congruent d-dimensional closed balls. The spindle convex body is called a…

This paper proves the volume of the arithmetic Okounkov body, constructed from a hermitian line bundle on an arithmetic variety by the author in a previous paper, is equal to the the volume of the hermitian line bundle up to a simple…

We derive an explicit formula for the volume of a regular simplex in the hyperbolic space of any dimension.

The Busemann-Petty problem asks whether origin-symmetric convex bodies in real Euclidean n-space with smaller central hyperplane sections necessarily have smaller volume. The answer is affirmative for n less or equal to 4 and negative if n…

If (M^n, g) is a complete Riemannian manifold with filling radius at least R, then we prove that it contains a ball of radius R and volume at least c(n)R^n. If (M^n, hyp) is a closed hyperbolic manifold and if g is another metric on M with…

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…

This thesis consists of five papers about reduced spherical convex bodies and in particular spherical bodies of constant width on the $d$-dimensional sphere $S^d$. In paper I we present some facts describing the shape of reduced bodies of…

We consider $k$-dimensional central sections of the unit ball of $\ell_p^n$ (denoted $B_p^n$) and we prove that their volume are bounded by the volume of $B_p^n$ whenever $1<p<2$ and $1\le k\le (n-1)/2$ or $k=n-1$. We also consider $0<p<1$…

In this paper an explicit formula for a lower bound on the volume of a hyperbolic orbifold, dependent on dimension and the maximal order of torsion in the orbifolds' fundamental group, is constructed.

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…

We generalize the hyperplane inequality in dimensions up to 4 to the setting of arbitrary measures in place of the volume. To prove this generalization we establish stability in the affirmative part of the solution to the Busemann-Petty…

Let $K$ be an $n$-dimensional convex body. Define the difference body by $$ K-K= \{x-y \mid x,y \in K \}. $$ We estimate the volume of the section of $K-K$ by a linear subspace $F$ via the maximal volume of sections of $K$ parallel to $F$.…

{\bf Abstract.} Let $D$ be a convex body of diameter $\delta$, where $0 < \delta < \frac{\pi}{2}$, on the $d$-dimensional sphere. We prove that $D$ is of constant diameter $\delta$ if and only if it is of constant width $\delta$ in the…

We study the volume of unit balls $B^n_{p,q}$ of finite-dimensional Lorentz sequence spaces $\ell_{p,q}^n.$ We give an iterative formula for ${\rm vol}(B^n_{p,q})$ for the weak Lebesgue spaces with $q=\infty$ and explicit formulas for $q=1$…