Let $K$ be a convex body in $\mathbb R^d$ and let $\theta \in (0,\infty)$.

>**Question.** *Is it always possible to find $n$ points $x_1,\dotsc,x_n \in \mathbb R^d$ such that
$$
\theta K \subseteq \operatorname{conv}(x_1,\dotsc,x_n) \subseteq K
\tag{1}\label{1}
$$
 and $n \le C_\theta d$ for some constant $C_\theta < \infty$ which only depends on $\theta$?*

I'm interested in the case where $\theta \to 1$.

Some known results:
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- From **Theorem 1.1** of "[Approximating a convex body by a polytope using the epsilon-net theorem](https://arxiv.org/abs/1705.07754)", we know that if $\theta=1/d$, then $n = 500d$ points drawn uniformly at random of from $K$ satisfy \eqref{1} with probability $1-e^{-d}$. The issue with this result is that it only works for $\theta \ll 1$. 

- From **Theorem 1.2** of the same paper, we know that if $n=_{\theta} \mathcal O(\exp(d))$ are drawn uniformly at random from $K$, then \eqref{1} is satisfied with probability $1-e^{-d}$. The issue with this result is that the dependence of $n$ on $d$ is exponential. We need linear.

On second thought it seems the bounds implied by the above results are an artifact of the complete randomness in the points $x_1,\dotsc,x_n$.