On Isoperimetric Stability

  title={On Isoperimetric Stability},
  author={Vsevolod F. Lev},
  journal={Discrete Analysis},
  • V. Lev
  • Published 16 September 2017
  • Mathematics
  • Discrete Analysis
We show that a non-empty subset of an abelian group with a small edge boundary must be large; in particular, if $A$ and $S$ are finite, non-empty subsets of an abelian group such that $S$ is independent, and the edge boundary of $A$ with respect to $S$ does not exceed $(1-\gamma)|S||A|$ with a real $\gamma\in(0,1]$, then $|A| \ge 4^{(1-1/d)\gamma |S|}$, where $d$ is the smallest order of an element of $S$. Here the constant $4$ is best possible. As a corollary, we derive an upper bound for the… Expand


Edge-Isoperimetric Problem for Cayley Graphs and Generalized Takagi Functions
  • V. Lev
  • Mathematics, Computer Science
  • SIAM J. Discret. Math.
  • 2015
For homocyclic groups G, the number of edges from A to its complement G in the directed Cayley graph is denoted by $\partial_S(A)$, which is an explicit closed-form expression in the case where A is an initial segment of $G$ with respect to the lexicographic order induced by S. Expand
Sets with large additive energy and symmetric sets
We show that for any set A in a finite Abelian group G that has at least c|A|^3 solutions to a"1+a"2=a"3+a"4, a"[email protected]?A there exist sets A^'@?A and @[email protected]?G,Expand
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  • David Reimer
  • Computer Science, Mathematics
  • Combinatorics, Probability and Computing
  • 2003
It is proved that, if a collection of sets [Ascr ] is union-closed, then the average set size of A is at least $\frac{1}{2}\log_2 (\vert A}\vert)$. Expand
An average set size theorem, Combinatorics, Probability and Computing
  • 2003