Kusner关于等边集合的一个问题

平面巴拿赫-马祖尔紧致集的等边维数

Equilateral dimension of the planar Banach--Mazur compactum

本文证明了在巴拿赫-马祖尔距离下，平面对称凸体中存在任意大的等边集，并展示了构造的最优性。

Equilateral sets and a Schütte Theorem for the 4-norm

A well-known theorem of Schütte (1963) gives a sharp lower bound for the ratio between the maximum distance and minimum distance between n+2 points in n-dimensional Euclidean space. In this brief note we adapt Bárány's elegant proof of this theorem to the space $\ell_4^n$. This gives a new proof that the largest cardinality of an equilateral set in $\ell_4^n$ is n+1, and gives a constructive bound for an interval $(4-ε_n,4+ε_n)$ of values of p close to 4 for which it is guaranteed that the largest cardinality of an equilateral set in $\ell_p^n$ is n+1.

Equilateral sets in the $\ell_1$ sum of Euclidean spaces

Let $E^n$ denote the (real) $n$-dimensional Euclidean space. It is not known whether an equilateral set in the $\ell_1$ sum of $E^a$ and $E^b$, denoted here as $E^a \oplus_1 E^b$, has maximum size at least $\dim(E^a \oplus_1 E^b) + 1 = a + b + 1$ for all pairs of $a$ and $b$. We show, via some explicit constructions of equilateral sets, that this holds for all $a \leqslant 27$, as well as some other instances.

Around the Petty theorem on equilateral sets

The main goal of this paper is to provide an alternative proof of the following theorem of Petty: in the normed space of dimension at least three, every 3-element equilateral set can be extended to a 4-element equilateral set. Our approach is based on the result of Kramer and Németh about inscribing a simplex into a convex body. To prove the theorem of Petty, we shall also establish that for every 3 points in the normed plane, forming an equilateral set of the common distance $p$, there exists a fourth point, which is equidistant to the given points with the distance not larger than $p$. We wi…

Tilings of convex sets by mutually incongruent equilateral triangles contain arbitrarily small tiles

We show that every tiling of a convex set in the Euclidean plane $\mathbb{R}^2$ by equilateral triangles of mutually different sizes contains arbitrarily small tiles. The proof is purely elementary up to the discussion of one family of tilings of the full plane $\mathbb{R}^2$, which is based on a surprising connection to a random walk on a directed graph.

Large equilateral sets in subspaces of $\ell_\infty^n$ of small codimension

For fixed $k$ we prove exponential lower bounds on the equilateral number of subspaces of $\ell_{\infty}^n$ of codimension $k$. In particular, we show that if the unit ball of a normed space of dimension $n$ is a centrally symmetric polytope with at most $\frac{4n}{3}-o(n)$ pairs of facets, then it has an equilateral set of cardinality at least $n+1$. These include subspaces of codimension $2$ of $\ell_{\infty}^{n+2}$ for $n\geq 9$ and of codimension $3$ of $\ell_{\infty}^{n+3}$ for $n\geq 15$.

Maximal equilateral sets

A subset of a normed space $X$ is called equilateral if the distance between any two points is the same. Let $m(X)$ be the smallest possible size of an equilateral subset of $X$ maximal with respect to inclusion. We first observe that Petty's construction of a $d$-dimensional $X$ of any finite dimension $d\geq 4$ with $m(X)=4$ can be generalised to give $m(X\oplus_1\mathbb{R})=4$ for any $X$ of dimension at least 2 which has a smooth point on its unit sphere. By a construction involving Hadamard matrices we then show that for any set $Γ$, $m(\ell_p(Γ))$ is finite and bounded above by a functio…

ℓ_p^n中超过n+1的等边集：线性过剩与维度代价

Equilateral sets in $\ell_p^n$ beyond $n+1$: Linear excess and dimension cost

该研究探讨ℓ_p^n中等边集的最大点数问题，证明p>4时过剩点数至少随维度线性增长，p=4为边界，相关结果在Lean中形式化。

三维对跖集与模等边集

Three-dimensional antipodal and norm-equilateral sets

我们刻画了至少允许六个或至少七个等距点的三维空间。特别地，我们证明了$\R^3$上允许六个等距点的$C^\infty$范数的存在性，这反驳了Lawlor和Morgan（1994，Pacific J. Math \textbf{166}，55- 83）的一个猜想，并给出了$\R^3$上某些一致凸范数的具有六个区域的能量最小化锥的存在性.另一方面，$\R^3$上的可微范数不允许有7个等距点。证明中的一个关键要素是对所有三维对跖集的分类。我们还将所得结果应用于几个三维凸体的接触数。

Similar dissection of sets

In 1994, Martin Gardner stated a set of questions concerning the dissection of a square or an equilateral triangle in three similar parts. Meanwhile, Gardner's questions have been generalized and some of them are already solved. In the present paper, we solve more of his questions and treat them in a much more general context. Let $D\subset \mathbb{R}^d$ be a given set and let $f_1,...,f_k$ be injective continuous mappings. Does there exist a set $X$ such that $D = X \cup f_1(X) \cup ... \cup f_k(X)$ is satisfied with a non-overlapping union? We prove that such a set $X$ exists for certain cho…

双色平面上的单色三角形

Monochromatic triangles in two-colored plane

我们证明了，对于任何分区的平面成一个闭集$C$和一个开集$O$和任何配置$T$的三个点，有一个平移和旋转的副本$T$包含在$C$或在$O$。除此之外，我们考虑分区的平面分为两组，其共同边界是一个联盟的分段线性曲线。我们表明，对于任何这样的分区和任何配置$T$这是一个顶点集的一个非等边三角形有一个副本的$T$包含在内部的两个分区类。此外，我们给出了这些“多边形”分区，避免一个给定的等边三重副本的特征。这些结果支持了Erdos、Graham、Montgomery、Rothschild、Spencer和Straus的猜想，该猜想指出平面的每一个二色都包含一个非等边点三元组的单色副本;另一方面，我们通过提供避免给定等边点三元组的二色的非平凡例子，反驳了同一作者的一个更强的猜想。

曼哈顿度量下的指数级奇距离集

Exponential odd-distance sets under the Manhattan metric

本文在曼哈顿度量下构造 $\mathbb{R}^n$ 中规模为 $2^n$ 的奇距离集，改进已有线性下界，并证明整数或半整数坐标限制下的最优性。

Lebesgue通用覆盖问题中Brass--Sharifi下界的可复现证明

A Certified Lower Bound for Lebesgue's Universal Cover Problem

通过结合几何估计与计算机搜索，Brass和Sharifi证明了Lebesgue通用覆盖问题凸形式的下界0.832；本文给出了该计算基于证书的复现，包含有限自适应账本、终端路径重放、三个局部下界证书族、大型表格的紧凑完整性审计以及将重放数据与下界陈述连接的有限证明义务层。