immediate from the definition that the function $f$ has Lipschitz constant $s$.

Note that $f(\vy) = h(\vy)$ for $\vy \in {\mathbb{H}_{d}}$ and $f(\vx) \geq h({\texttt{round}}(\vx))$ for all $\vx \in {B_d(0, R)}$. This implies that the minimum value of $f$ is the same as the minimum value of $h$, and indeed any ($\eps$-)minimizer $\vy$ of $h$ also ($\eps$-)minimizes $f$. Conversely, let $\vx$ be an $\eps$-minimizer of $f$. Since $h({\texttt{round}}(\vx)) \leq f(\vx)$, it follows that ${\texttt{round}}(\vx)$ is an $\eps$-minimizer of $h$.

Finally we prove the equivalence of approximate sampling. Towards that goal, we define an intermediate distribution on ${B_d(0, R)}$. Let $\widehat{{{\mathcal{D}}_f^{\lambda}}}$ be the distribution ${{\mathcal{D}}_f^{\lambda}}$ conditioned on being in $\cup_{\vy \in {\mathbb{H}_{d}}} {B_d(\vy, \frac 1 4)}$.

We first argue that $\eta$-samplability of $\widehat{{{\mathcal{D}}_f^{\lambda}}}$ is equivalent to $\eta$-samplability of ${\mathcal{D}}_h^{\lambda, {\mathbb{H}_{d}}}$. Suppose that $X$ is a sample from $\widehat{{{\mathcal{D}}_f^{\lambda}}}$. Then for any $\vy^\star \in {\mathbb{H}_{d}}$, $$\begin{aligned}
\Pr[X \in {B_d(\vy^\star, \frac 1 4)}] &= \frac{\int_{{B_d(\vy^\star, \frac 1 4)}} \exp(-\lambda f(\vx))\;{\mathrm{d}}\vx}{ \sum_{\vy \in {\mathbb{H}_{d}}}\int_{{B_d(\vy, \frac 1 4)}} \exp(-\lambda f(\vx))\;{\mathrm{d}}\vx}\\
&= \frac{\exp(-\lambda h(\vy^\star)) \cdot \int_{{B_d(\vy^\star, \frac 1 4)}} \exp(-{s}\lambda g(\vx)) \;{\mathrm{d}}\vx}{\sum_{\vy \in {\mathbb{H}_{d}}} \exp(-\lambda h(\vy)) \cdot \int_{{B_d(\vy, \frac 1 4)}} \exp(-{s}\lambda g(\vx)) \;{\mathrm{d}}\vx}\\
&= \frac{\exp(-\lambda h(\vy^\star))}{\sum_{\vy \in {\mathbb{H}_{d}}}\exp(-\lambda h(\vy))}\end{aligned}$$ Thus ${\texttt{round}}(X)$ is a sample from 