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 $γ\colon [0,1]\to\mathcal{X}$, there exists a $1$-Lipschitz function $F\colon \mathcal{X}\to \mathbb{R}$ that catches the total variation of $γ$, i.e. such that $\operatorname{V}_{F\circγ} \gtrsim \operatorname{V}_γ$. 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 $\mathbb{E} |f(x) - f(y)| \gtrsim ρ(x,y)$ for every $x$ and $y$ in $\mathcal{X}$.

Disclosure

“i=0 this implies the existence of the desired 1-Lipschitz function f such that Vf ◦γ ≳ Vγ . The reverse implication presented in this note is slightly subtler. Acknowledgement. The author acknowledges the use of ChatGPT (OpenAI) during the development of the proof. All mathematical arguments were checked and are the responsibility of the author. 2 Main theorem Denote the Lipschitz constant of a mapping f by Lip f . By a 1-Lipschitz function we always mean a”

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Classification

Proof ideas or individual proof-step assistance

Multiplier

8

Verified

2026-08-23T05:08:21.885558Z

Structural counts