[Renner and Wolf 2004] introduces the notion of $\epsilon$-smooth Renyi entropies as $$H_\alpha^\epsilon(P) \equiv \frac{1}{1-\alpha} \inf_{Q\in \mathcal B^\epsilon(P)}\log\left(\sum_z Q(z)^\alpha\right),$$ for $\alpha\in[0,\infty]$, $\epsilon\ge 0$, and $\mathcal B^\epsilon(P) = \{Q: \, \delta(P,Q)\le \epsilon\}$ is the set of probability distributions $\epsilon$-close to $P$ in total variation distance $\delta$.
In the paper, they then go on to define the "$\epsilon$-extractable uniform randomness" of a set $\mathcal P$ of probability distributions as the "amount of randomness that can be extracted from a random variable $Z$ with any probability distribution $P_Z\in\mathcal P$": $$H_{\rm ext}^\epsilon(\mathcal P) \equiv \max_U \log |U|$$ maximised over uniform random variables $U$ such that for some random function $F$, $(U,F)$ is $\epsilon$-close to $(F(Z),F)$ for any random variable $Z$.
The paper goes on to claim that $H_{\rm ext}^\epsilon(\mathcal P)$ is quantified by the smooth Renyi entropy for $\alpha\to\infty$.
Is there any simple example where one can compute explicitly smooth entropies and extractable uniform randomness to see this relation explicitly?
