# Better lower bound on min-entropy

In “Randomness Condensers for Efficiently Samplable, Seed-Dependent Sources” by Dodis, Ristenpart, and Vadhan (PDF), I have seen the statement that:

any tuple of distributions $$(X,Z)$$ is $$\varepsilon$$-close to $$(X′, Z)$$ such that $$H_\infty(X′\mid Z) ≥ H_2(X\mid Z) − \log (1/\varepsilon)$$ offering a bound on min-entropy which may be better than $$H_\infty(X′\mid Z) ≥ H_2(X\mid Z)/2$$.

I can not see why this is the case. Could anyone please shed some light? Does it involve the Leftover Hash Lemma!?

For some $0<\epsilon<1$, let $B$ be the subset of heaviest points in $X$ whose mass add up to $\epsilon$ -- i.e.: $$\sum_{x\in B} p_X(x)=\epsilon, \text{ and } \min_{x\in B}p_X(x)\geq\max_{x\in\bar{B}}p_X(x).$$ Let $X'$ denote the conditional distribution of $X$ on $\bar{B}$ -- i.e.: $$p_{X'}(x):= \begin{cases} 0 & x\in B,\\ \frac{p_{X}(x)}{(1-\epsilon)} & x\in\bar{B}. \end{cases}$$ Note that the statistical distance between $X$ and $X'$, $\Delta(X,X')=\epsilon$.$^1$ Let $m:=\max_{x\in\bar{B}}p_X(x)$. Now, let's consider the collision entropy of $X$: $$H_2(X):=-\log\sum_{x\in X} p_X^2(x)\leq-\log\sum_{x\in B} p_X^2(x)\leq^2-\log (m\epsilon).$$ It follows that $-\log(m)\geq H_2(X)-\log(1/\epsilon)$, and the results follows as $H_\infty(X')=-\log(m/(1-\epsilon))$.
Footnotes: 1. Proof: \begin{align} \Delta(X,X') &=\frac{1}{2}\cdot\sum_{x\in X}\left|p_X(x)-p_{X'}(x)\right|\\ &=\frac{1}{2}\cdot\sum_{x\in B}\left|p_X(x)\right|+\sum_{x\in \bar{B}}\left|p_X(x)-p_{X'}(x)\right|\\ &=\frac{1}{2}\cdot\left(\epsilon+ \sum_{x\in \bar{B}}p_X(x)\left|1-\frac{1}{1-\epsilon}\right|\right)\\ &=\frac{1}{2}\cdot\left(\epsilon+ (1-\epsilon)\left|1-\frac{1}{1-\epsilon}\right|\right)=\epsilon \end{align} 2. Proof: $\sum_{x\in B} p_X^2(x)\geq \sum_{x\in B}m\cdot p_X(x)= m\cdot\epsilon$