Let $X$ be a set, $\mathcal{M}(X)$ denote the set of metrics on $X$, and fix a $\mathcal{F}\subseteq \mathbb{R}^{\mathcal{X}}$. Let $L \in\mathbb{R},\delta\ge 0$. For any metric $\rho$ on $X$, we say that a map $f:X\to \mathbb{R}$ is $(L,\delta)$-quasi-Lipschitz if: for every $x,y\in X$ $$ |f(x)-f(y)|\le L\,\rho(x,y)+\delta. $$ Let $\operatorname{Lip}((X,\rho)|L,\delta)$ denote the set of all $(L,\delta)$-quasi-Lipschitz maps on $(X,\rho)$.
For any fixed $\delta\ge 0$. We define the $\delta$-quasi-Lipschitz upper-envelope to be infimum $$ M_{\delta}:=\inf\big\{L\ge 1:\, (\exists \rho\in \mathcal{M}(X))\, \operatorname{Lip}((X,\rho)|1,\delta) \subseteq \mathcal{F} \subseteq \operatorname{Lip}((X,\rho)|L,\delta)\big\}. $$ That is, $M_{\delta}$ represents the *best embedding of the function class $\mathcal{F}$ into a class of $1$-Lipschitz maps; note that I consider $1$-Lipschitz maps since, without loss of generality, one can rescale the said metric by a constant.
Intuitively, I am asking if every set of functions is almost the set of ``normalized'' Lipschitz maps for some metric, up to some discretization/quantization error $\delta\ge 0$.
My question is:
- Are the reasonable conditions on $\mathcal{F}$ under which $M_{\delta}$ is finite and can be ensured to be "small"?
- If $\rho$ is a metric realizing $M_{\delta}$, can we say something about its metric dimension?
My intuition is that $M_{\delta}$ should be related to the pseudodimension of Pollard, where
he pseudodimension of $\mathcal{F}$, written $\operatorname{Pdim}(\mathcal{F})$, is the largest integer $D$ for which there exists
$(x_1, \ldots, x_D, y_1, \ldots, y_D) \in X^D \times \mathbb{R}^D$ such that for any $(b_1, \ldots, b_m) \in \{0,1\}^D$ there exists $f \in \mathcal{F}$ such that for all $i=1,\dots,D$
$$
\quad f(x_i) > y_i \Leftrightarrow b_i = 1.
$$
