度量几何
Zauner's conjecture concerns the existence of $d^2$ equiangular lines in $\mathbb{C}^d$; such a system of lines is known as a SIC. In this paper, we construct infinitely many new SICs over finite fields. While all previously known SICs…
Let $N$ denote the maximum number of congruent infinite cylinders that can be arranged in $\mathbb{R}^3$ so that every pair of cylinders touches each other. Littlewood posed the question of whether $N=7$, which remains unsolved. In this…
A Voronoi diagram partitions the plane into convex cells, each containing the points closest to a single generator. Given such a tessellation, the inverse Voronoi problem seeks the generator set \( S \) that produced it. Our algorithm…
Let $M_{n, m}(\mathbb{R})$ be the space of $n\times m$ real matrices. Define $\mathcal{K}_o^{n,m}$ as the set of convex compact subsets in $M_{n,m}(\mathbb{R})$ with nonempty interior containing the origin $o\in M_{n, m}(\mathbb{R})$, and…
In this paper, we generalise an interesting geometry problem from the 1995 edition of the International Mathematical Olympiad (IMO) using analytic geometry tools.
A shape of a combinatorial polytope is a convex embedding into Euclidean space. We provide necessary and sufficient conditions for a piecewise linear map between two shapes of the same polytope to be a compression (respectively a weak…
In this work we prove the Stepanov differentiation theorem for multiple-valued functions. This theorem is proved in the wide generality of metric-space-multiple-valued functions without relying on a Lipschitz extension result. General…
The so-called "einstein problem" (a pun playing with the famous scientist's name and the German term "ein Stein" for "one stone") asks for a simply connected prototile only allowing nonperiodic tilings without need of any matching rule. So…
We propose a definition of non-collapsed space with Ricci curvature bounded from below and we prove the versions of Colding's volume convergence theorem and of Cheeger-Colding dimension gap estimate for ${\sf RCD}$ spaces. In particular…
We establish dimension-free stability of Webb's sharp simplex slicing (1996). Incidentally, we investigate Lipschitzness of volume of hyperplane central sections of arbitrary (not necessarily symmetric) convex bodies.
The expected number of real projective roots of orthogonally invariant random homogeneous real polynomial systems is known to be equal to the square root of the B\'ezout number. A similar result is known for random multi-homogeneous…
Inscribed angles are investigated in taxicab geometry with application to the existence and uniqueness of inscribed and circumscribed taxicab circles of triangles.
While the concept of straight-line length is well understood in taxicab geometry, little research has been done into the length of curves or the nature of area and volume in this geometry. This paper sets forth a comprehensive view of the…
A natural analogue to angles and trigonometry is developed in taxicab geometry. This structure is then analyzed to see which, if any, congruent triangle relations hold. A nice application involving the use of parallax to determine the exact…
Hierarchical graphs often exhibit tree-like branching patterns, a structural property that challenges the design of traditional graph filters. We introduce a boundary-weighted operator that rescales each edge according to how far its…
In Euclidean geometry, the Pythagorean theorem is presented as an equation involving three squares. This paper explores how analogous expressions may be identified in spherical and hyperbolic geometries.
Motivated by the discrete logarithmic Minkowski problem we study for a given matrix $U\in\mathbb{R}^{n\times m}$ its cone-volume set $C_{\tt cv}(U)$ consisting of all the cone-volume vectors of polytopes $P(U,b)=\{ x\in\mathbb{R}^n :…
In this survey, we introduce a new curvature-dimension condition for extended metric measure spaces, called Barycenter-Curvature Dimension condition BCD, from the perspective of Wasserstein barycenter.
We study the Wasserstein barycenter problem in the setting of non-compact, non-smooth extended metric measure spaces. We introduce a couple of new concepts and obtain the existence, uniqueness, absolute continuity of the Wasserstein…
In a recently published article by G. Ambrus et al. a new \emph{upper bound} for the density of an unit avoiding, periodic set is given as $0.2470$, the first upper bound $< 1/4$. A construction of Croft 1967 gave a \emph{lower bound}…