The distance problem on measured metric spaces

Abstract

What distributions arise as the distribution of the distance between two typical points in some measured metric space? This seems to be a surprisingly subtle problem. We conjecture that every distribution with a density function whose support contains $0$ does arise in this way, and give some partial results in that direction.

Figures (2)

• Figure 1: Illustrating construction of the tree structure $T = (T,\rho,d,\nu)$ for $n = 3$.
• Figure 2: A part of the construction of $(S,d,\mu)$. Let $\xi_1$ and $\xi_2$ be independent random variables with distribution $\mu$, and let $u_1\neq u_2$ be leaves of $T$. Conditionally on ${\xi_1\in X^{u_1}}$ and ${\xi_2\in X^{u_2}}$, the random variable $d(\xi_1,\xi_2)$ has the same distribution as ${\varepsilon/n\cdot(1+U_1)+d(u_1,u_2)+\varepsilon/n\cdot(1+U_2)=d(u_1,u_2)+\varepsilon/n\cdot(2+U_1+U_2)}$, where $U_1$ and $U_2$ are independent random variables with uniform distribution on $[0,1]$. Moreover, by construction, we have $d(u_1,u_2)=d_i+j\cdot\varepsilon\cdot n^{-2}$ for some $i\in\llbracket1,k\rrbracket$ and $j\in J_n$.

Theorems & Definitions (19)

• Proposition 1
• Proposition 2
• Theorem 1
• Proposition 3
• Corollary 1
• proof : Proof of Proposition \ref{['prop:0supp']}
• Remark 1
• Proposition 4
• proof
• Proposition 5