Metric Geometry Papers of the Week

We prove the existence of a sharp threshold $p_0 \approx 26.265\ldots$, such that for every $n\in\mathbb{N}$ and $p\geq p_0$, the volume maximizing section of the unit ball of $\ell_p^n$ is the section with Ball's hyperplane $(\frac{e_1+e_2}{\sqrt{2}})^\perp$, whereas for $p\in(2,p_0)$ the asymptotic maximizer as the ambient dimension $n\to\infty$ is the hyperplane orthogonal to the diagonal vector $\frac{e_1+\cdots+e_n}{\sqrt{n}}$.

We study intersections of $K$ with rectifiable curves and chord-arc curves.

We prove, in every dimension, the upper bound conjectured by Betke, Henk and Wills for the lattice-point enumerator in terms of volume and successive minima.

We prove that every convex body $K\subset\mathbb{R}^n$ admits $S\in\mathrm{SL}_n(\mathbb{R})$ such that $\lambda_1(SK)|K|^{2/n}\le Cn$, where $C$ is universal.

In this paper, we develop a general theory of escape-covering duality for any dimension and explore several applications.

We introduce the concept of Markov-driven self-similar measures, a notion that simultaneously generalizes classical self-similar measures, self-conformal measures, dynamically driven self-similar measures and many random measures.

We build on a recent proof of the universal optimality of the planar hexagonal lattice.

On a volume doubling metric measure space endowed with a family of $p$-energies such that the Poincaré inequality and the capacity upper bound with $p$-walk dimension $\beta_p$ hold, for $p$ in an open interval $I\subseteq(1,+\infty)$, we prove the following dichotomy: either $\beta_p=p$ for all $p\in I$, or the function $p\mapsto \frac{\beta_p}{p}$ takes values in $(1,+\infty)$ and is strictly decreasing on $I$.

We give three ways to quantify the distance from flagness, and in each case we bound the diameter in a way that interpolates between the Adiprasito-Benedetti and the Larman-Barnette bounds.

We prove that standard $k$-bubbles uniquely minimize perimeter in $\mathbb R^n$ and $\mathbb S^n$ for $n\ge2$ and $1\le k\le n$, and that standard double bubbles uniquely minimize perimeter in $\mathbb H^n$ for $n\ge3$.

We prove a homotopical Euclidean isoperimetric inequality for Lipschitz maps from the unit n-sphere into metric spaces admitting a bounded bicombing, which includes CAT(0)-spaces and normed vector spaces.

We study the corresponding notion for curves, where point averages are replaced by normalized arclength integration.

This paper introduces a class of non-homogeneous self-similar constructions, called $\omega $-substitution graph system, in which the interface vertices are determined by an infinite sequence $\omega .

We show that any $n$-point Euclidean metric admits a non-expansive embedding into the line with average distortion at most $O\left(\sqrt{\frac{\log n}{\log\log n}}\right)$, matching a known lower bound.

We construct a compact set of diameter $1$ in $\mathbb{R}^5$ which cannot be covered by six Euclidean balls of diameter $1$.

Under the usual nondegeneracy assumptions, repeatedly projecting a fixed point onto the side lines of an $n$-gon returns its shape after $n$ steps. We explain this classical theorem through a short complex-number identity: the multipliers of one step move cyclically around the polygon.

We give an elementary proof of a tangent-cone identity which might be known to specialists in geometric measure theory.

We prove that every lattice isomorphism between the lattices of slowly oscillating real-valued functions on proper metric spaces induces a coarse homeomorphism.

We characterize neutral planes by three metric conditions: existence of geodesics, side-side-side congruence, and uniqueness of reflection.

In this paper, we continue our investigation of the structure of Thurston geometries with non-constant curvature.