度量几何

Although the Sierpi\'nski triangle has planar area $0$, it is uniformly non-flat: at every point and every scale, its nearby points span a two-dimensional region of comparable size. We prove a sharp version of this statement, showing that…

This paper introduces a natural definition for the volume of the unit ball in $n$-dimensional normed spaces $\mathbb{R}^n$. This definition preserves the Euclidean relation $P(B)/V(B)=n$ between the perimiter and the volume of the unit ball…

An N-tiling of triangle ABC by triangle T is a way of writing ABC as a union of N triangles congruent to T, overlapping only at their boundaries. The triangle T is the "tile". The tile may or may not be similar to ABC. In this paper we…

We prove that if $C$ is a symmetric convex body of revolution in $\mathbb R^4$ containing the unit Euclidean ball $\mathbb B_4$, such that the sections of $C$ by hyperplanes tangent to $\mathbb B_4$ have constant area $A>0$, then $C$ is a…

A variant of the flatness problem from integer programming is studied, in which one considers convex bodies in $\mathbb{R}^d$ with at most $k$ interior lattice points. The maximum lattice width of such a body is denoted by Flt(d,k) and it…

Let us consider a sphere $S^{n-1}$ of radius $r$ in $\mathbb{R}^n$, where we have fixed poles $N$ and $S$. Suppose that $K$ is a set in $\mathbb{R}^n$ containing a translated copy of each meridian (that is an $S^{n-2}$-sphere) of $S^{n-1}$.…

We prove that over a generic Poncelet triangle family, the locus of the circumcenter of an inversive triangle is a conic. Additionally, we prove an earlier conjecture: over generic Poncelet triangles, two unique points exist which maintain…

Any function from a round $n$-dimensional sphere of radius $r$ into $n$-dimensional Euclidean space must distort the metric additively by at least $\displaystyle \frac{\pi r}{1 + \sqrt{1 - \frac{2}{n+2}}}$ if $n$ is even and $\displaystyle…

The illumination number $I(K)$ of a convex body $K$ in Euclidean space $\mathbb{E}^d$ is the smallest number of directions that completely illuminate the boundary of a convex body. A cap body $K_c$ of a ball is the convex hull of a…

A convex polyhedron is Rupert if a hole can be cut into it (making its genus $1$) such that an identical copy of the polyhedron can pass through the hole. Resolving a conjecture of Jerrard-Wetzel-Yuan, Steininger and Yurkevich recently…

The medial axis $M_X$ of a closed set $X\subset \mathbb{R}^n$ is the set of points from the ambient space that admit more than one closest point in $X$. We study the problem of reaching the singularities, i.e. of characterising the points…

The orthogonal beltway problem is the problem of recovering the $\mathrm{O}(n)$-orbit of a $\delta$-function supported at a finite number of points in $\r^n$ from its auto-correlation or, equivalently, second moment. It was introduced as a…

The medial axis of a closed set is well established tool in pattern recognition, cherished for its power of reconstruction of domains. In this article we fill this gap answering the question which sets precisely are reconstructible from the…

We prove that an infinitesimally Hilbertian CD(0,N) space containing a line splits as the product of $R$ and an infinitesimally Hilbertian CD(0,N-1) space. By `infinitesimally Hilbertian' we mean that the Sobolev space $W^{1,2}(X,d,m)$,…

In 2016, Viazovska famously solved the sphere packing problem in dimension $8$, using modular forms to construct a 'magic' function satisfying optimality conditions determined by Cohn and Elkies in 2003. In March 2024, Hariharan and…

The main goal of this paper is to address the following problem: given a positive integer $n$, find the largest value $S(n)$ such that a square of edge length $S(n)$ in the Euclidean plane can be covered by $n$ unit squares. We investigate…

In this paper, we obtain the limit formula of the observable diameter with non-Euclidean screen. In order to treat a sequence of observable diameters with varying screens, we define new types of observable diameters with errors.

What is the largest constant $c\in [0,1]$ with the property that every finite collection $\mathcal{C}$ of axis-parallel squares in the plane admits a disjoint sub-collection $\mathcal{S}$ occupying at least a fraction $c$ of the area…

We prove polarity duality for covering problems in Hilbert geometry. Let $G$ and $K$ be convex bodies in $\mathbb{R}^d$ where $G \subset \operatorname{int}(K)$ and $\operatorname{int}(G)$ contains the origin. Let $N^H_K(G,\alpha)$ and…

In 2014 the author showed that in the three-dimensional spherical space, alongside with three classical types of flexible octahedra constructed by Bricard, there exists a new type of flexible octahedra, which was called exotic. In the…