Approximate convexity and an edge-isoperimetric estimate

Abstract

We study extremal properties of the function $$ F(x) := \min\{k\|x\|^{1-1/k}\colon k\ge 1\},\ x\in[0,1], $$ where $\|x\|=\min\{x,1-x\}$. In particular, we show that $F$ is the pointwise largest function of the class of all real-valued functions $f$ defined on the interval $[0,1]$, and satisfying the relaxed convexity condition $$ f(tx_1+(1-t)x_2) \le tf(x_1)+(1-t)f(x_2)+|x_2-x_1|, \ x_1,x_2,t\in[0,1] $$ and the boundary condition $\max\{f(0),f(1)\}\le 0$. As an application, we prove that if $A$ and $S$ are subsets of a finite abelian group $G$, such that $S$ is generating and all of its elements have order at most $m$, then the number of edges from $A$ to its complement $G\setminus A$ in the directed Cayley graph induced by $S$ on $G$ is $$ \partial_S(A) \ge \frac{1}{m} |G| F(|A|/|G|). $$