首页/论文/1010.1990

arXiv 1010.19902010-10-11math.MG

An elementary proof of borsuk theorem

Dian Yang

备注：: 2 pages, 1 figure, this note is one of the results of my participation in `Math in Moscow' program

英文摘要：

In 1933, Borsuk conjectured that any bounded d-dimensional set of nonzero diameter can be broken into d + 1 parts of smaller diameter. This conjecture was disproved for large enough d, though it is true for low dimensional cases. The paper provides an alternative proof for d = 2 case.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

A Computer Program for Borsuk's Conjecture

In 1933, Borsuk proposed the following problem: Can every bounded set in $\mathbb{E}^n$ be divided into $n+1$ subsets of smaller diameters? This problem has been studied by many authors, and a lot of partial results have been discovered. In particular, Kahn and Kalai's counterexamples surprised the mathematical community in 1993. Nevertheless, the problem is still far away from being completely resolved. This paper presents a broad review on related subjects and, based on a novel reformulation, introduces a computer proof program to deal with this well-known problem.

2020-01-11math.MG

正交群的Borsuk-Ulam型定理与正交四分割

Borsuk-Ulam type theorem for the orthogonal group and orthogonal four-partitions

本文给出Makeev提出的正交四分割猜想的首个完整证明，核心是证明正交群上的Borsuk-Ulam型定理，通过构造显式模型映射并利用等变度原理完成证明。

2026-09-21math.ATmath.COmath.MG

通过Reuleaux多面体研究Borsuk问题与Vázsonyi问题

Borsuk and Vázsonyi problems through Reuleaux polyhedra

本文利用Reuleaux多面体建立了三维空间中Borsuk数为4的临界集与Vázsonyi问题极小结构之间的等价关系，并由此完整刻画了所有Borsuk数为4的有限集合。

2025-07-22math.COmath.MG

彩色 Borsuk–Ulam 定理及其应用

Colorful Borsuk--Ulam theorems and applications

本文建立 Borsuk–Ulam 定理的彩色及高对称推广，并据此推出火腿三明治、Fan 球面覆盖、Gale 彩色 KKM 与 Brouwer 不动点等定理的彩色形式。

2023-09-25math.COmath.MG

63维Borsuk问题的一个AI生成反例

An AI Generated Counterexample to Borsuk Problem in Dimension 63

该研究由ChatGPT利用GPT 5.6 Sol构造出63维含321个点的集合，证明b(63)≥65，为Borsuk问题提供了AI生成的反例。

2026-08-17math.COmath.MG

Beyond the Borsuk-Ulam theorem: The topological Tverberg story

Bárány's "topological Tverberg conjecture" from 1976 states that any continuous map of an $N$-simplex $Δ_N$ to $\mathbb{R}^d$, for $N\ge(d+1)(r-1)$, maps points from $r$ disjoint faces in $Δ_N$ to the same point in $\mathbb{R}^d$. The proof of this result for the case when $r$ is a prime, as well as some colored version of the same result, using the results of Borsuk-Ulam and Dold on the non-existence of equivariant maps between spaces with a free group action, were main topics of Matoušek's 2003 book "Using the Borsuk-Ulam theorem." In this paper we show how advanced equivariant topology meth…

2017-05-21math.ATmath.COmath.MG

更多相关论文收起更多论文

A 64-dimensional two-distance counterexample to Borsuk's conjecture

In 1933 Karol Borsuk asked whether each bounded set in the n-dimensional Euclidean space can be divided into n+1 parts of smaller diameter. The diameter of a set is defined as the supremum (least upper bound) of the distances of contained points. Implicitly, the whole set is assumed to contain at least two points. The hypothesis that the answer to that question is positive became famous under the name Borsuk's conjecture. Beginning with Jeff Kahn and Gil Kalai, from 1993 to 2003 several authors have proved that in certain (almost all) high dimensions such a division is not generally possible. …

2014-08-20math.COmath.MG

Metric Thickenings, Borsuk-Ulam Theorems, and Orbitopes

Thickenings of a metric space capture local geometric properties of the space. Here we exhibit applications of lower bounding the topology of thickenings of the circle and more generally the sphere. We explain interconnections with the geometry of circle actions on Euclidean space, the structure of zeros of trigonometric polynomials, and theorems of Borsuk-Ulam type. We use the combinatorial and geometric structure of the convex hull of orbits of circle actions on Euclidean space to give geometric proofs of the homotopy type of metric thickenings of the circle. Homotopical connectivity bounds …

2019-10-09math.ATmath.COmath.GT

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…

2021-03-28math.FAmath.MG

An Elementary Proof of the 3 Dimensional Simplex Mean Width Conjecture

After a Hessian computation, we quickly prove the 3D simplex mean width conjecture using classical methods. Then, we generalize some components to $d$ dimensions.

2021-08-08cs.ITmath.ITmath.MG