• Corpus ID: 231718845

# The cylindrical width of transitive sets

@inproceedings{Sah2021TheCW,
title={The cylindrical width of transitive sets},
author={Ashwin Sah and Mehtaab Sawhney and Yufei Zhao},
year={2021}
}
• Published 27 January 2021
• Mathematics
The following counterintuitive fact was conjectured by the third author and proved by Green [4]. It says that every finite transitive subset of a high dimensional sphere is close to some hyperplane. Here a subset X of a sphere in Rd is transitive if for every x, x′ ∈ X, there is some g ∈ O(Rd) so that gX = X and gx = x′. We say that X has width at most 2r if it lies within distance r of some hyperplane. The finiteness assumption is important since otherwise the whole sphere is a counterexample.

## References

SHOWING 1-10 OF 11 REFERENCES
On Jordan's theorem for complex linear groups
Abstract In 1878, Jordan showed that a finite subgroup of GL(n, ℂ) must possess an abelian normal subgroup whose index is bounded by a function of n alone. We will give the optimal bound for all n;
On the width of transitive sets: Bounds on matrix coefficients of finite groups
We say that a finite subset of the unit sphere in $\mathbf{R}^d$ is transitive if there is a group of isometries which acts transitively on it. We show that the width of any transitive set is bounded
Recent developments in non-asymptotic theory of random matrices
In these notes, the techniques aimed for obtaining explicit probability bounds in non-asymptotic theory of random matrices are described and some recent results are surveyed.
High-Dimensional Probability: An Introduction with Applications in Data Science
© 2018, Cambridge University Press Let us summarize our findings. A random projection of a set T in R n onto an m-dimensional subspace approximately preserves the geometry of T if m ⪆ d ( T ) . For...
An introduction with applications in data science, With a foreword
• Statistical and Probabilistic Mathematics,
• 2018
Restricted invertibility revisited, A journey through discrete mathematics
• 2017
A nonasymptotic theory of independence
• Concentration inequalities
• 2013
Concentration inequalities, Oxford University Press, Oxford, 2013, A nonasymptotic theory of independence, With a foreword
• 2013
Multiplicative number theory, third ed
• Graduate Texts in Mathematics,
• 2000