A quantity measuring the separability of Banach spaces retort

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A quantity measuring the separability of Banach spaces

Let $X$ breathe a Banach space. It is unaffected for us to insert a quantity measuring the separability of sets as follows: for a subset $A$ of $X$, we clique

$textrm{sep}(A)=inf{epsilon>0: Asubseteq K+epsilon B_{X}$ for some countable subset $K$ of $X}$.

Clearly, $A$ is separable if and only if $textrm{sep}(A)=0$.

It is elementary that a Banach space $X$ is separable if $X^{*}$ is separable. My question is to give a quantitative version of this known outcome.

Question. $textrm{sep}(B_{X})leq Ccdot textrm{sep}(B_{X^{*}})$ for some universal ceaseless $C$ ?

Thank you.

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