Related papers: On a Simplex Inscribed in a Ball

In the spirit of the Genetics of the Regular Figures, by L. Fejes T\'oth, we prove the following theorem: If $2n$ points are selected in the $n$-dimensional Euclidean ball $B^n$ so that the smallest distance between any two of them is as…

We show that a realization of a closed connected PL-manifold of dimension n-1 in n-dimensional Euclidean space (n>2) is the boundary of a convex polyhedron (finite or infinite) if and only if the interior of each (n-3)-face has a point,…

Let $C$ be a convex body and let $S$ be a nondegenerate simplex in ${\mathbb R}^n$. Denote by $\xi(C;S)$ the minimal $\tau>0$ such that $C$ is a subset of the simplex $\tau S$. By $\alpha(C;S)$ we mean the minimal $\tau>0$ such that $C$ is…

Given a degenerate $(n+1)$-simplex in a $d$-dimensional space $M^d$ (Euclidean, spherical or hyperbolic space, and $d\geq n$), for each $k$, $1\leq k\leq n$, Radon's theorem induces a partition of the set of $k$-faces into two subsets. We…

Let $B$ be a Euclidean ball in ${\mathbb R}^n$ and let $C(B)$ be a space of~continuous functions $f:B\to{\mathbb R}$ with the uniform norm $\|f\|_{C(B)}:=\max_{x\in B}|f(x)|.$ By $\Pi_1\left({\mathbb R}^n\right)$ we mean a set of…

Let $M^d$ be the spherical, Euclidean, or hyperbolic space of dimension $d\ge n+1$. Given any degenerate $(n+1)$-simplex $\mathbf{A}$ in $M^d$ with non-degenerate $n$-faces $F_i$, there is a natural partition of the set of $n$-faces into…

Let $\De$ be a non-degenerate simplex on $k$ vertices. We prove that there exists a threshold $s_k<k$ such that any set $A\subs \R^k$ of Hausdorff dimension $dim\,A\geq s_k$ necessarily contains a similar copy of the simplex $\De$.

Determining the minimum density of a covering of $\mathbb{R}^{n}$ by Euclidean unit balls as $n\to\infty$ is a major open problem, with the best known results being the lower bound of $\left(\mathrm{e}^{-3/2}+o(1)\right)n$ by Coxeter, Few…

We derive lower estimates for the approximation of the $d$-dimensional Euclidean ball by polytopes with a fixed number of $k$-dimensional faces, $k\in\{0,1,\ldots,d-1\}$. The metrics considered include the intrinsic volume difference and…

We prove that there is an absolute constant $ C$ such that for every $ n \geq 2 $ and $ N\geq 10^n, $ there exists a polytope $ P_{n,N} \subset \mathbb{R}^n $ with at most $ N $ facets that satisfies…

Let $M$ be a smooth, connected, compact submanifold of $\mathbb{R}^n$ without boundary and of dimension $k\geq 2$. Let $\mathbb{S}^k \subset \mathbb{R}^{k+1}\subset \mathbb{R}^n$ denote the $k$-dimesnional unit sphere. We show if $M$ has…

Let $K \subset \mathbb R^n$ be a convex body with barycenter at the origin. We show there is a simplex $S \subset K$ having also barycenter at the origin such that $\left(\frac{vol(S)}{vol(K)}\right)^{1/n} \geq \frac{c}{\sqrt{n}},$ where…

The investigation of the relation among the distances of an arbitrary point in the Euclidean space $\mathbb{R}^n$ to the vertices of a regular $n$-simplex in that space has led us to the study of simplices having a regular facet. Calling an…

In this paper we prove that a flat free-boundary minimal $n$-disk, $n\geq3$, in the unit Euclidean ball $B^{n+1}$ is the unique compact free boundary minimal hypersurface in the unit Euclidean ball which the squared norm of the second…

Let F be the fundamental group of S, where S is a compact, connected, oriented surface with negative Euler characteristic and nonempty boundary. (1) The projective class of the chain \partial S in B_1(F) intersects the interior of a…

In this work we prove the following: let $K$ be a convex body in the Euclidean space $\mathbb{R}^n$, $n\geq 3$, contained in the interior of the unit ball of $\mathbb{R}^n$, and let $p\in \mathbb{R}^n$ be a point such that, from each point…

We construct geometric barriers for minimal graphs in H^n xR. We prove the existence and uniqueness of a solution of the vertical minimal equation in the interior of a convex polyhedron in H^n extending continuously to the interior of each…

M. Goresky and R. MacPherson intersection homology is also defined from the singular chain complex of a filtered space by H. King, with a key formula to make selections among singular simplexes. This formula needs a notion of dimension for…

We show that if $M^n$ is a properly immersed, two-sided, stable minimal hypersurface in $B^{n+1}_1(0)\setminus S$, where $S$ is closed with $\mathcal{H}^{n-2}(S)=0$, then $\text{dim}_{\mathcal{H}}\text{sing}(M)\leq n-7$, namely…

A simple n-gon is a polygon with n edges such that each vertex belongs to exactly two edges and every other point belongs to at most one edge. Brass, Moser and Pach asked the following question: For n > 3 odd, what is the maximum perimeter…