Approximate convexity and an edge-isoperimetric estimate

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:416

Article

Lev, Vsevolod F.

关键词: Approximate convexity;    Edge-isoperimetric inequalities;   

DOI  :  10.1016/j.jmaa.2014.02.049

来源: Elsevier

【 摘 要 】

We study extremal properties of the function F(x) := min{k parallel to x parallel to(1-1/k) ; k >= 1}, x epsilon [0,1], where parallel to x parallel to =min{x,1-x}. In particular, we show that F is the pointwise largestfunction of the class of all real-valued functions f defined on the interval [0,1], and satisfying the relaxed convexity condition f(lambda x(1)+(1-lambda)x(2)) <= lambda f(x(1))+(1-lambda)f(x(2))+ vertical bar x(2)-x(1 vertical bar), x(1),x(2),lambda epsilon [0,1] and the boundary condition max{f(0),f(1)} <= 0. As an application, we prove that if A and S are subsets of a finiteabelian group $G$, such that $S$ is generating and all of its elements haveorder at most $m$, then the number of edges from A to its complement$Gsetminus A$ in the directed Cayley graph induced by S on G is partial derivative S(A) >= 1/m vertical bar G vertical bar F(vertical bar A vertical bar/vertical bar G vertical bar). (c) 2014 Elsevier Inc. All rights reserved.

【 授权许可】

Free