Related papers: Low $M^*$-estimates on coordinate subspaces

The longstanding Godbersen's conjecture states that for any convex body $K \subset \mathbb R^n$ of volume $1$ and any $j \in \{0, \ldots, n\}$, the mixed volume $V_j = V(K[j], -K[n - j])$ is bounded by $\binom{n}{j}$, with equality if and…

Let $K$ and $L$ be two convex bodies in ${\mathbb R^5}$ with countably many diameters, such that their projections onto all $4$ dimensional subspaces containing one fixed diameter are directly congruent. We show that if these projections…

Suppose $D$ is a suitably admissible compact subset of $\mathbb{R}^k$ having a smooth boundary with possible zones of zero curvature. Let \mbox{$R(T,\theta,x)= N(T,\theta,x) - T^{k}\mathrm{vol}(D)$,} where $N(T,\theta,x)$ is the number of…

We prove several estimates for the moments of arbitrary measures on convex bodies. We apply these estimates to show a new slicing inequality for measures on convex bodies. We also deduce estimates for the outer volume ratio distance from an…

We consider an isoperimetric inequality for $(m+1)$-dimensional area minimizing submanifolds of arbitrary codimension which lie outside a convex set $\mathcal{K} \subset \mathbb{R}^{n+1}$ and are bounded by a submanifold of…

Equivalence in physics is discussed on the basis of experimental data accompanied by experimental errors. The introduction of the equivalence being consistent with the mathematical definition is possible only in theories constructed on…

In this paper we show error bounds for randomly subsampled rank-1 lattices. We pay particular attention to the ratio of the size of the subset to the size of the initial lattice, which is decisive for the computational complexity. In the…

We strongly believe that in order to prove two important geometrical pro\-blems in convexity, namely, the G. Bianchi and P. Gruber's Conjecture \cite{bigru} and the J. A. Barker and D. G. Larman's Conjecture \cite{Barker}, it is necessary…

In this paper, with the aid of the simplicial approximation property, the Hopf's construction and Dugundji's homotopy extension Theorem, we first show that if C is a nonempty compact convex subset of an F-space (E; || ||); then for every…

A long-standing conjecture states that all normalized symplectic capacities coincide on the class of convex subsets of ${\mathbb R}^{2n}$. In this note we focus on an asymptotic (in the dimension) version of this conjecture, and show that…

Coverings of convex bodies have emerged as a central component in the design of efficient solutions to approximation problems involving convex bodies. Intuitively, given a convex body $K$ and $\epsilon> 0$, a covering is a collection of…

In this work we bring together tools and ideology from two different fields, Symplectic Geometry and Asymptotic Geometric Analysis, to arrive at some new results. Our main result is a dimension-independent bound for the symplectic capacity…

In a previous work we proved that each $n$-dimensional convex polyhedron ${\mathcal K}subset{\mathbb R}^n$ and its relative interior are regular images of ${\mathbb R}^n$. As the image of a non-constant polynomial map is an unbounded…

Let $s\in (0,1)$, and let $F\subset \mathbb{R}$ be a self similar set such that $0 < \dim_H F \leq s$ . We prove that there exists $\delta= \delta(s) >0$ such that if $F$ admits an affine embedding into a homogeneous self similar set $E$…

It is proved that if $C$ is a convex body in ${\Bbb R}^n$ then $C$ has an affine image $\widetilde C$ (of non-zero volume) so that if $P$ is any 1-codimensional orthogonal projection, $$|P\widetilde C| \ge |\widetilde C|^{n-1\over n}.$$ It…

Let $\Lambda \subset \mathbb R^n$ be an algebraic lattice, coming from a projective module over the ring of integers of a number field $K$. Let $\mathcal Z \subset \mathbb R^n$ be the zero locus of a finite collection of polynomials such…

Let $\Delta=\Delta_1\times\ldots\times \Delta_d\subseteq\mathbb{R}^n$, where $\mathbb{R}^n=\mathbb{R}^{n_1}\times\cdots\times\mathbb{R}^{n_d}$ with each $\Delta_i\subseteq\mathbb{R}^{n_i}$ a non-degenerate simplex of $n_i$ points. We prove…

In this work we prove that if for a pair of convex bodies $K_1, K_2 \subset \mathbb{R}^n$, $n \geq 3$, there exists a hyperplane $H$ and two distinct points $p_1$ and $p_2$ in $\mathbb{R}^n \setminus H$ such that for every $(n-2)$-plane $M…

We propose convex optimization algorithms to recover a good approximation of a point measure $\mu$ on the unit sphere $S\subseteq \mathbb{R}^n$ from its moments with respect to a set of real-valued functions $f_1,\dots, f_m$. Given a finite…

In this paper we motivate some new directions of research regarding the lattice width of convex bodies. We show that convex bodies of sufficiently large width contain a unimodular copy of a standard simplex. This implies that every lattice…