on $f$ on $\Re^d$ such that optimization of $f$ and $h$ are reducible to each other, and similarly sampling from $f$ and $h$ are reducible to each other. This would allow us to use separation results for the hypercube case to establish analagous separation results for the continuous case.

![image](figures/function_h.pdf){width=".48\textwidth"} ![image](figures/function_f.pdf){width=".48\textwidth"}

\[thm:discrete\_to\_cont\_v1\] Let $h : {\mathbb{H}_{d}} \rightarrow \Re$ have range $[0, d]$. Fix $M \geq 2d, R \geq 2\sqrt{d}$. Then there is a funtion $f : {B_d(0, R)}\rightarrow \Re$ satisfying the following properties:

Efficiency:

:   Given $\vx \in {B_d(0, R)}$ and oracle access to $h$, $f$ can computed in polynomial time.

Lipschitzness:

:   $f$ is continuous and $L$-Lipschitz for $L= 2M$.

Sampler Equivalence:

:   Fix $\lambda \geq \frac{4d \ln 24R}{M}$. Given access to an $\eta$-sampler for ${\mathcal{D}}_h^{\lambda, {\mathbb{H}_{d}}}$, there is an efficient $\eta'$-sampler for ${{\mathcal{D}}_f^{\lambda}}$, for $\eta' = \eta + \exp(-\Omega(d))$. Conversely, given access to an $\eta$-sampler for ${{\mathcal{D}}_f^{\lambda}}$, there is an efficient $\eta'$-sampler for ${\mathcal{D}}_h^{\lambda, {\mathbb{H}_{d}}}$ for $\eta' = \eta + \exp(-\Omega(d))$.

The function $f$ is fairly natural: it takes a large value $M \geq 2d$ at most points, except in small balls around the hypercube vertices. At each hypercube vertex, $f$ is equal to the $h$ value at 