度量几何
The 'popcorn function' isThe `popcorn function' is a well-known and important example in real analysis with many interesting features. We prove that the box dimension of the graph of the popcorn function is 4/3, as well as computing the…
We prove a general nonlinear projection theorem for Assouad dimension. This theorem has several applications including to distance sets, radial projections, and sum-product phenomena. In the setting of distance sets we are able to…
This note extends the invariant defined in "An invariant of metric spaces under bornologous equivalences" to the coarse category.
We use properties of the hyperbolic metric and properties of the modular function to show that the Bohr's radius for covering maps onto hyperbolic domains is greater or equal to exponential minus pi. This includes almost all known classes…
The article presents the mathematical sequences describing circle packing densities in four different geometric configurations involving a hexagonal lattice based equal circle packing in the Euclidian plane. The calculated sequences take…
We consider 2-step free-Carnot groups equipped with sub-Finsler distances. We prove that the metric spheres are codimension-one rectifiable from the Euclidean viewpoint. The result is obtained by studying how the Lipschitz constant for the…
We generalize a bi-Lipschitz extension result of David and Semmes from Euclidean spaces to complete metric measure spaces with controlled geometry (Ahlfors regularity and supporting a Poincar\'e inequality). In particular, we find sharp…
The Brunn-Minkowski theory in convex geometry concerns, among other things, the volumes, mixed volumes, and surface area measures of convex bodies. We study generalizations of these concepts to Borel measures with density in…
We explore some inequalities in convex geometry restricted to the class of zonoids. We show the equivalence, in the class of zonoids, between a local Alexandrov-Fenchel inequality, a local Loomis-Whitney inequality, the log-submodularity of…
We represent minimal upper gradients of Newtonian functions, in the range $1\le p<\infty$, by maximal directional derivatives along "generic" curves passing through a given point, using plan-modulus duality and disintegration techniques. As…
Based on a notion by Gray and Kambites of hyperbolicity in the setting of semimetric spaces like digraphs or semigroups, we will construct (under a small additional geometric assumption) a boundary based on quasi-geodesic rays and anti-rays…
Gray and Kambites introduced a notion of hyperbolicity in the setting of semimetric spaces like digraphs or semigroups. We will prove that under a small additional geometric assumption their notion of hyperbolicity is preserved by…
The celebrated Johnson-Lindenstrauss lemma states that for all $\varepsilon \in (0,1)$ and finite sets $X \subseteq \mathbb{R}^N$ with $n>1$ elements, there exists a matrix $\Phi \in \mathbb{R}^{m \times N}$ with…
After calculating the Dushnik-Miller dimension of Minkowski spaces to be countable infinity, we define a novel notion of dimension for ordered spaces recovering the correct manifold dimension and obtain a corresponding obstruction for the…
The purpose of this paper is to introduce the new concept of weighted floating functions associated with log concave or $s$-concave functions. This leads to new notions of weighted functional affine surface areas. Their relation to more…
The classical Steinitz theorem states that if the origin belongs to the interior of the convex hull of a set $S \subset \mathbb{R}^d$, then there are at most $2d$ points of $S$ whose convex hull contains the origin in the interior.…
These notes derive aperiodic monotiles (arXiv:2303.10798) from a set of rhombuses with matching rules. This dual construction is used to simplify the proof of aperiodicity by considering the tiling as a colouring game on a Rhombille tiling.…
In this article, we prove that for a definable set in an o-minimal structure with connected link (at 0 or infinity), the inner distance of the link is equivalent to the inner distance of the set restricted to the link. With this result, we…
This paper deals with the problem of reconstructing the path of a vehicle in an unknown environment consisting of planar structures using sound. Many systems in the literature do this by using a loudspeaker and microphones mounted on a…
Letting $P$ be a convex polytope in $\mathbb{R}^d$ with $n>d$ vertices, we study geometric and analytical properties of the set of generalized barycentric coordinates relative to any point $p\in P$. We prove that such sets are polytopes in…