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Yanlu Lian

Publications and source records attributed to Yanlu Lian.

10 recordsLinked to original sources
arXiv2026

Non-Separable Homothetic Triangles, Part I: Constructions and Lower Bounds

A finite family of planar convex bodies is called a non-separable family if no line disjoint from its union has at least one member in each open half-plane. In this paper, we prove that there exist finite non-separable families of positive homothetic triangles with covering factors strictly exceeding the sharp three-member bound $\mu = \frac{2}{3} + \frac{2}{3\sqrt{3}}$. This resolves negatively a question posed by K. Bezdek and Z. L\'angi, who originally established this three-member bound after proving that the classic factor 1 covering theorem by A. W. Goodman and R. E. Goodman for disks fails for arbitrary positive homothets. Besides giving explicit algebraic examples with four, five, and six members having factors of approximately 1.0533161, 1.0551900, and 1.0572061 respectively, we provide a common cyclic recurrence that yields a 303-member family with the exact factor 250000000/235141779. Finally, we derive a continuous model suggested by increasingly fine recurrences, yielding a numerical candidate of 1.0633083; its attainability and optimality remain open.

math.MG↗
arXiv2026

Minimal Covering Bodies and a Minkowski-Type Criterion for Lattice Coverings

The structural characterization of lattice coverings is a fundamental problem in the geometry of numbers. In particular, a covering analogue to Minkowski's criterion for lattice packings has remained open. In this paper, we introduce the concept of \textit{minimal covering bodies} and investigate their structural properties. First, we establish a lattice covering criterion in three dimensions based on the Kuhn triangulation. Furthermore, while three-dimensional parallelohedra admit only finitely many combinatorial types, we prove the existence of infinitely many combinatorial types of minimal covering bodies in both the three-dimensional asymmetric case and the four-dimensional centrally symmetric case. Finally, we propose a Minkowski-type geometric criterion and an algebraic intersection framework, which reduce the three-dimensional covering problem to a finite computational verification.

math.MG↗
arXiv2024

The Tammes Problem in $\mathbb{R}^{n}$ and Linear Programming Method

The Tammes problem delves into the optimal arrangement of $N$ points on the surface of the $n$-dimensional unit sphere (denoted as $\mathbb{S}^{n-1}$), aiming to maximize the minimum distance between any two points. In this paper, we articulate the sufficient conditions requisite for attaining the optimal value of the Tammes problem for arbitrary $n, N \in \mathbb{N}^{+}$, employing the linear programming framework pioneered by Delsarte et al. Furthermore, we showcase several illustrative examples across various dimensions $n$ and select values of $N$ that yield optimal configurations. The findings illuminate the intricate structure of optimal point distributions on spheres, thereby enriching the existing body of research in this domain.

math.MG↗
arXiv2023

On Landau-Kato inequalities via semigroup orbits

Let $\omega>0$. Given a strongly continuous semigroup $\{e^{tA}\}$ on a Banach space and an element $f\in\mathbf{D}(A^2)$ satisfying the exponential orbital estimates $$\|e^{tA}f\|\leq e^{-\omega t}\|f\| \quad\text{and}\quad \|e^{tA}A^2f\|\leq e^{-\omega t}\|A^2f\|,\quad t\geq0,$$ a dynamical inequality for $\|Af\|$ in terms of $\|f\|$ and $\|A^2f\|$ was derived by Herzog and Kunstmann (Studia Math., 2014). Here we provide an improvement of their result by relaxing the exponential decay to quadratic, together with a simple and direct way recovering the usual Landau inequality. Herzog and Kunstmann also demanded an analogue, again via semigroup orbits, for the Kato type inequality on Hilbert spaces. We provide such a result by using Hayashi-Ozawa machinery [Proc. Amer. Math. Soc., (2017)] which in turn relies on Hilbertian geometry.

math.FA↗
arXiv2023

Lower Bound on Translative Covering Density of Octahedron

In this paper, we present the first nontrivial lower bound on the translative covering density of octahedron. To this end, we show the lower bound, in any translative covering of octahedron, on the density relative to a given parallelehedron. The resulting lower bound on the translative covering density of octahedron is $1+6.6\times10^{-8}$.

math.MG↗
arXiv2021

On Hadwiger's covering functional for the simplex and the cross-polytope

In 1957, Hadwiger made a conjecture that every $n$-dimensional convex body can be covered by $2^n$ translations of its interior. The Hadwiger's covering functional $γ_m(K)$ is the smallest positive number $r$ such that $K$ can be covered by $m$ translations of $rK$. Due to Zong's program, we study the Hadwiger's covering functional for the simplex and the cross-polytope. In this paper, we give upper bounds for the Hadwiger's covering functional of the simplex and the cross-polytope.

math.MG↗
arXiv2021

Divide bounded sets into sets having smaller diameters

For each positive integer $m$ and each real finite dimensional Banach space $X$, we set $β(X,m)$ to be the infimum of $δ\in (0,1]$ such that each set $A\subset X$ having diameter $1$ can be represented as the union of $m$ subsets of $A$ whose diameters are at most $δ$. Elementary properties of $β(X,m)$, including its stability with respect to $X$ in the sense of Banach-Mazur metric, are presented. Two methods for estimating $β(X,m)$ are introduced. The first one estimates $β(X,m)$ using the knowledge of $β(Y,m)$, where $Y$ is a Banach space sufficiently close to $X$. The second estimation uses the information about $β_X(K,m)$, the infimum of $δ\in(0,1]$ such that $K\subset X$ is the union of $m$ subsets having diameters not greater than $δ$ times the diameter of $K$, for certain classes of convex bodies $K$ in $X$. In particular, we show that $β(l_p^3,8)\leq 0.925$ holds for each $p\in [1,+\infty]$ by applying the first method, and we proved that $β(X,8)<1$ whenever $X$ is a three-dimensional Banach space satisfying $β_X(B_X,8)<\frac{221}{328}$, where $B_X$ is the unit ball of $X$, by applying the second method. These results and methods are closely related to the extension of Borsuk's problem in finite dimensional Banach spaces and to C. Zong's computer program for Borsuk's conjecture.

math.FA↗
arXiv2021

Covering the crosspolytope with its smaller homothetic copies

In 1957, Hadwiger made the famous conjecture that any convex body of $n$-dimensional Euclidean space $\mathbb{E}^n$ can be covered by $2^n$ smaller positive homothetic copies. Up to now, this conjecture is still open for all $n\geq 3$. Denote by $γ_{m}(K)$ the smallest positive number $λ$ such that $K$ can be covered by $m$ translations of $λK$. The values of $γ_m(K)$ for some particular $m$ and $K$ have been studied. In this article, we will focus on the situation where $K$ is the unit crosspolytope of the three-dimensional.

math.MG↗