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What do we know about the covering number of $L$-Lipschitz functions mapping say, $\mathbb{R}^n \rightarrow \mathbb{R}$ for some $L >0$?

Only 2 results I have found so far are,

• That the $\infty$-norm covering number for $L$-Lipschitz functions constrained to map $[0,1]^d \rightarrow [0,1]$ is $\exp\left(\Theta\left( L/\epsilon\right)^d\right)$. And for this I could not find a reference for the proof.

• Another such $\infty$-norm covering number count for $1$-Lipschitz functions mapping an unit diameter metric space to $[-1,1]$ was given in this previously unanswered question here, metric entropy for Lipschitz functions

  Even here I can't find the proof anywhere and the link in the MO question seesm to go to something else (and the original reference in the Uspekhi Mat. Nauk, 1959, Volume 14, Issue 2(86), Pages 3–86, is in Russian and hence I can't read it).

Is there some reason why we don't have (is provably impossible?) such counts when either the range or the domain of the function space is unbounded?

It would be great if someone can maybe reference the proofs for the two results quoted here!

What do we know about the covering number of $L$-Lipschitz functions mapping say, $\mathbb{R}^n \rightarrow \mathbb{R}$ for some $L >0$?

Only 2 results I have found so far are,

• That the $\infty$-norm covering number for $L$-Lipschitz functions constrained to map $[0,1]^d \rightarrow [0,1]$ is $\exp\left(\Theta\left( L/\epsilon\right)^d\right)$. And for this I could not find a reference for the proof.

• Another such $\infty$-norm covering number count for $1$-Lipschitz functions mapping an unit diameter metric space to $[-1,1]$ was given in this previously unanswered question here, metric entropy for Lipschitz functions

  Even here I can't find the proof anywhere and the link in the MO question seems to go to something else (and the original reference in the Uspekhi Mat. Nauk, 1959, Volume 14, Issue 2(86), Pages 3–86, is in Russian and hence I can't read it).

Is there some reason why we don't have (is provably impossible?) such counts when either the range or the domain of the function space is unbounded?

It would be great if someone can maybe reference the proofs for the two results quoted here!

What do we know about the covering number of $L-$Lipschitz functions mapping say, $\mathbb{R}^n \rightarrow \mathbb{R}$ for some $L >0$?

Only 2 results I have found so far are,

• That the $\infty-$norm covering number for $L-$Lipschitz functions constrained to map $[0,1]^d \rightarrow [0,1]$ is $e^{\Theta (\left ( \frac{L}{\epsilon} \right )^d)}$. And for this I could not find a reference for the proof.

• Another such $\infty-$norm covering number count for $1-$Lipschitz functions mapping an unit diameter metric space to $[-1,1]$ was given in this previously unanswered question here, metric entropy for Lipschitz functions

  Even here I cant find the proof anywhere and the link in the MO question seesm to go to something else. (and the original reference in the Uspekhi Mat. Nauk, 1959, Volume 14, Issue 2(86), Pages 3–86, is in Russian and hence I can't read)

Is there some reason why we dont have (is provably impossible?) such counts when either the range or the domain of the function space is unbounded?

It would be great if someone can maybe reference the proofs for the two results quoted here!

What do we know about the covering number of $L$-Lipschitz functions mapping say, $\mathbb{R}^n \rightarrow \mathbb{R}$ for some $L >0$?

Only 2 results I have found so far are,

• That the $\infty$-norm covering number for $L$-Lipschitz functions constrained to map $[0,1]^d \rightarrow [0,1]$ is $\exp\left(\Theta\left( L/\epsilon\right)^d\right)$. And for this I could not find a reference for the proof.

• Another such $\infty$-norm covering number count for $1$-Lipschitz functions mapping an unit diameter metric space to $[-1,1]$ was given in this previously unanswered question here, metric entropy for Lipschitz functions

  Even here I can't find the proof anywhere and the link in the MO question seesm to go to something else (and the original reference in the Uspekhi Mat. Nauk, 1959, Volume 14, Issue 2(86), Pages 3–86, is in Russian and hence I can't read it).

Is there some reason why we don't have (is provably impossible?) such counts when either the range or the domain of the function space is unbounded?

It would be great if someone can maybe reference the proofs for the two results quoted here!

What do we know about the covering number of $L-$Lipschitz functions mapping say, $\mathbb{R}^n \rightarrow \mathbb{R}$ for some $L >0$?

Only 2 results I have found so far are,

• That the $\infty-$norm covering number for $L-$Lipschitz functions constrained to map $[0,1]^d \rightarrow [0,1]$ is $e^{\Theta (\left ( \frac{L}{\epsilon} \right )^d)}$. And for this I could not find a proof.

• Another such $\infty-$norm covering number count for $1-$Lipschitz functions mapping an unit diameter metric space to $[-1,1]$ was given in this previously unanswered question here, metric entropy for Lipschitz functions

  Even here I cant find the proof anywhere and the link there is broken. (and the original reference in the Uspekhi Mat. Nauk, 1959, Volume 14, Issue 2(86), Pages 3–86, is in Russian and hence I can't read)

Is there some reason why we dont have (is provably impossible?) such counts when either the range or the domain of the function space is unbounded?

It would be great if someone can maybe reference the proofs for the two results quoted here!

What do we know about the covering number of $L-$Lipschitz functions mapping say, $\mathbb{R}^n \rightarrow \mathbb{R}$ for some $L >0$?

Only 2 results I have found so far are,

• That the $\infty-$norm covering number for $L-$Lipschitz functions constrained to map $[0,1]^d \rightarrow [0,1]$ is $e^{\Theta (\left ( \frac{L}{\epsilon} \right )^d)}$. And for this I could not find a reference for the proof.

• Another such $\infty-$norm covering number count for $1-$Lipschitz functions mapping an unit diameter metric space to $[-1,1]$ was given in this previously unanswered question here, metric entropy for Lipschitz functions

  Even here I cant find the proof anywhere and the link in the MO question seesm to go to something else. (and the original reference in the Uspekhi Mat. Nauk, 1959, Volume 14, Issue 2(86), Pages 3–86, is in Russian and hence I can't read)

Is there some reason why we dont have (is provably impossible?) such counts when either the range or the domain of the function space is unbounded?

It would be great if someone can maybe reference the proofs for the two results quoted here!