English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

In 1933, Karol Borsuk asked whether each bounded set in the $n$-dimensionalEuclidean space can be divided into $n$+1 parts of smaller diameter. Because itwould not make sense otherwise, one usually assumes that he just forgot torequire that the whole set contains at least two points. The hypothesis thatthe answer to that question is positive became famous under the name\emph{Borsuk's conjecture}. Counterexamples are known for any $n\ge 64$, since2013.  Let $\Lambda$ the (original, unscaled) Leech lattice, a now very well-knowninfinite discrete vector set in the 24-dimensional Euclidean space. Thesmallest norm of nonzero vectors in $\Lambda$ is $\sqrt{32}$. Let $M$ the setof the 196560 vectors in $\Lambda$ having this norm.  For each $x \in M$, $-x$ is in $M$. Let $H$ the set of all subsets of $M$that for each $x$ in $M$ contain either $x$ or $-x$. Each element of $H$ hasthe same diameter $d = \sqrt{96}$. For dimensions $n<24$ one can analogouslyconstruct respective $M_n$ and $H_n$ from laminated $n$-dimensional sublatticesof $\Lambda$. For uniformity, let $\Lambda_{24}=\Lambda$, $M_{24} = M$ and$H_{24} = H$.  If $M_n$ is divisible into at most $n+1$ parts of diameter below $d$ thenthis applies to all elements of $H_n$, too. I have checked that this is thecase for all $n \le 21$. For $n$ from 22 to 24, the minimum number of parts ofdiameter below $d$ that I was able to divide $M_n$ into are 25, 29 and 35,resp. The source package of this article contains a data file encoding anelement of $H_{24}=H$ that I can not divide into less than 30 parts of smallerdiameter.

Sub-25-dimensional counterexamples to Borsuk's conjecture in the Leech lattice?