A random Lipschitz function detecting jumps

Abstract

Recently, A. Tyulenev and the author studied the class of metric spaces $\mathcal{X}$ such that for every mapping $\gamma :[0,1]\to \mathcal{X}$, there exists a $1$-Lipschitz function $F:\mathcal{X}\to \mathbb{R}$ that catches the total variation of $\gamma$, i.e. such that $V_{F\circ \gamma }\gtrsim V_{\gamma }$. In this short note, we show that a metric space $\mathcal{X}$ enjoys this property if and only if there exists a random $1$-Lipschitz function $f$ on $\mathcal{X}$ such that $\mathbb{E}|f(x)-f(y)|\gtrsim \rho (x,y)$ for every $x$ and $y$ in $\mathcal{X}$.

5 pages