Related papers: Upper bound for the Dvoretzky dimension in Milman-…

Let $0 \leq s \leq 1$. A set $K \subset \mathbb{R}^{2}$ is a Furstenberg $s$-set, if for every unit vector $e \in S^{1}$, some line $L_{e}$ parallel to $e$ satisfies $$\dim_{\mathrm{H}} [K \cap L_{e}] \geq s.$$ The Furstenberg set problem,…

Let $K$ be a convex body in $\mathbb{R}^n$ and $f : \partial K \rightarrow \mathbb{R}_+$ a continuous, strictly positive function with $\int\limits_{\partial K} f(x) d \mu_{\partial K}(x) = 1$. We give an upper bound for the approximation…

In this work we study subdivisions of $k$-rotationally symmetric planar convex bodies that minimize the maximum relative diameter functional. For some particular subdivisions called $k$-partitions, consisting of $k$ curves meeting in an…

Given a centrally symmetric convex body $K \subset \mathbb{R}^d$ and a positive number $\lambda$, we consider, among all ellipsoids $E \subset \mathbb{R}^d$ of volume $\lambda$, those that best approximate $K$ with respect to the symmetric…

The sphere packing problem asks for the greatest density of a packing of congruent balls in Euclidean space. The current best upper bound in all sufficiently high dimensions is due to Kabatiansky and Levenshtein in 1978. We revisit their…

Let $K\subset \mathbb{R}^n$ be a convex body, $n\geq 3$. We say that $K$ satisfies the Barker-Larman condition if there exists a ball $B$ in the interior of $K$ such that for every suppor hyperplane $\Pi$ of $B$, the section $\Pi \cap K$ is…

We show that the largest possible diameter $\delta(d,k)$ of a $d$-dimensional polytope whose vertices have integer coordinates ranging between $0$ and $k$ is at most $kd-\lceil2d/3\rceil$ when $k\geq3$. In addition, we show that…

We prove a weak version of the $\varepsilon$-Dvoretzky conjecture for normed spaces, showing the existence of a subspace of $\mathbb{R}^n$ of dimension at least $c \log n / |\log \varepsilon|$ in which the given norm is $\varepsilon$-close…

Denote by ${\mathcal K}^d$ the family of convex bodies in $E^d$ and by $w(C)$ the minimal width of $C \in {\mathcal K}^d$. We ask for the greatest number $\Lambda_n ({\mathcal K}^d)$ such that every $C \in {\mathcal K}^d$ contains a…

For every integer $k\geq 2$ and every $R>1$ one can find a dimension $n$ and construct a symmetric convex body $K\subset\mathbb{R}^n$ with $\text{diam}\,Q_{k-1}(K)\geq R\cdot\text{diam}\,Q_k(K)$, where $Q_k(K)$ denotes the $k$-convex hull…

In $\mathbb{R}^n$, we establish an asymptotically sharp upper bound for the upper Minkowski dimension of $k$-porous sets having holes of certain size near every point in $k$ orthogonal directions at all small scales. This bound tends to…

For a Minkowski centered convex compact set $K$ we define $\alpha(K)$ to be the smallest possible factor to cover $K \cap (-K)$ by a rescalation of $\mathrm{conv} (K\cup (-K))$ and give a complete description of the possible values of…

In this note we prove two results on the quantitative illumination parameter f(d) of the unit ball of a d-dimensional normed space introduced by K. Bezdek (1992). The first is that f(d) = O(2^d d^2 log d). The second involves Steiner…

The isodiametric inequality states that the Euclidean ball maximizes the volume among all convex bodies of a given diameter. We are motivated by a conjecture of Makai Jr.~on the reverse question: Every convex body has a linear image whose…

Given a subset K of the unit Euclidean sphere, we estimate the minimal number m = m(K) of hyperplanes that generate a uniform tessellation of K, in the sense that the fraction of the hyperplanes separating any pair x, y in K is nearly…

We study the slicing inequality for the surface area instead of volume. This is the question whether there exists a constant $\alpha_n$ depending (or not) on the dimension $n$ so that $$S(K)\leq\alpha_n|K|^{\frac{1}{n}}\max_{\xi\in…

We derive universal lower and upper bounds for max-min and min-max problems (also known as polarization) for the potential of spherical $(k,k)$-designs and provide certain examples, including unit-norm tight frames, that attain these…

We adapt an argument of Tao and Vu to show that if $\lambda_1\le\cdots\le\lambda_d$ are the successive minima of an origin-symmetric convex body $K$ with respect to some lattice $\Lambda<\mathbb{R}^d$, and if we set…

We study the complexity of geometric problems on spaces of low fractal dimension. It was recently shown by [Sidiropoulos & Sridhar, SoCG 2017] that several problems admit improved solutions when the input is a pointset in Euclidean space…

Here we show that any n-dimensional centrally symmetric convex body K has an n-dimensional perturbation T which is convex and centrally symmetric, such that the isotropic constant of T is universally bounded. T is close to K in the sense…