Related papers: The Archimedean Projection Property

For a convex body $K\subset\R^n$ and $i\in\{1,...,n-1\}$, the function assigning to any $i$-dimensional subspace $L$ of $\R^n$, the $i$-dimensional volume of the orthogonal projection of $K$ to $L$, is called the $i$-th projection function…

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,…

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…

Lebesgue measurable subsets A and B of parallel or identical k-dimensional affine subspaces of Euclidean n-space E^n satisfy The Product Formula for Volume: Vol_k(A)Vol_k(B) = \sum_{J \in S(n,k)} Vol_k({\pi}_J(A))Vol_k({\pi}_J(B)). Here…

In this paper we prove that $\Pi$-projective spaces $\mathbb{P}^n_\Pi$ arise naturally in supergeometry upon considering a non-projected thickening of $\mathbb{P}^n$ related to the cotangent sheaf $\Omega^1_{\mathbb{P}^n}$. In particular,…

We prove that for an arbitrary upper semi-continuous function $\phi\colon G(1,2) \to [0,1]$ there exists a compact set $F$ in the plane such that $\dim_{\textrm{A}} \pi F = \phi(\pi)$ for all $\pi \in G(1,2)$, where $\pi F$ is the…

We study the following geometry problem: given a $2^n-1$ dimensional vector $\pi=\{\pi_S\}_{S\subseteq [n], S\ne \emptyset}$, is there an object $T\subseteq\mathbb{R}^n$ such that $\log(\mathsf{vol}(T_S))= \pi_S$, for all $S\subseteq [n]$,…

This article deals with the existence of hypersurfaces minimizing general shape functionals under certain geometric constraints. We consider as admissible shapes orientable hypersurfaces satisfying a so-called reach condition, also known as…

We establish the following Hadamard--Stoker type theorem: Let $f:M^n\rightarrow\mathscr{H}^n\times\mathbb R$ be a complete connected hypersurface with positive definite second fundamental form, where $\mathscr H^n$ is a Hadamard manifold.…

We prove a restricted projection theorem for an n-2 dimensional family of projections from $\mathbb R^n$ to $\mathbb R$. The family we consider arises naturally in the context of the adjoint representation of the maximal unipotent subgroup…

This paper contains two results on the dimension and smoothness of radial projections of sets and measures in Euclidean spaces. To introduce the first one, assume that $E,K \subset \mathbb{R}^{2}$ are non-empty Borel sets with…

Let $(M,g_0)$ be a closed Riemannian manifold of dimension $n$, for $3 \leq n \leq 7$, and non-negative Ricci curvature. Let $g = \phi^2 g_0$ be a metric in the conformal class of $g_0$. We show that there exists a smooth closed embedded…

The Hardy space H^2(R) for the upper half plane together with a unimodular function group representation u(\lambda) = \exp(i(\lambda_1\psi_1 + ... + \lambda_n\psi_n)) for \lambda in R^n, gives rise to a manifold M of orthogonal projections…

We establish uniformization results for metric spaces that are homeomorphic to the euclidean plane or sphere and have locally finite Hausdorff 2-measure. Applying the geometric definition of quasiconformality, we give a necessary and…

We introduce and prove the $n$-dimensional Pizza Theorem: Let $\mathcal{H}$ be a hyperplane arrangement in $\mathbb{R}^{n}$. If $K$ is a measurable set of finite volume, the {pizza quantity} of $K$ is the alternating sum of the volumes of…

Let $K$ be a convex body in the 3-dimensional Euclidian space $\mathbb{E}^3$ and let $N,S$ in the boubdary bd$K$ of $K$, $N\not=S$. Suppose that the support plane $\Pi_S$ of $K$ at $S$ is unique. For every point $x$ in bd$ K$, different…

Extending results of Suss and Hadwiger (proved by them for the case of convex bodies and positive ratios), we show that compact (respectively, closed) convex sets in the Euclidean space of dimension n are homothetic provided for any given…

We derive upper bounds on the difference between the orthogonal projections of a smooth function $u$ onto two finite element spaces that are nearby, in the sense that the support of every shape function belonging to one but not both of the…

In projective dimension growth results, one bounds the number of rational points of height at most $H$ on an irreducible hypersurface in $\mathbb P^n$ of degree $d>3$ by $C(n)d^2 H^{n-1}(\log H)^{M(n)}$, where the quadratic dependence in…

A Riemannian n-manifold M has k-dimensional Uryson width bounded by a constant c >0 if there exists a continuous map f from M to an k-dimensional polyhedral space P, such that the pullbacks f^{-1}(p) of all points p in P have diameters…