Let $K\subset\mathbb R^n$ be a full-dimensional, bounded, convex polytope formed by the intersection of $m$ half-spaces. Is there a nice lower bound for the diameter $D(K) = \sup_{x,y\in K}\|x-y\|_2$ relative to a fixed volume $\text{Vol}(K)$?
Partial attempts
A simple lower bound is obtained via an "isodiametric inequality:" $$ \left( \frac{D(K)}{2} \right)^n \frac{\pi^{n/2}}{\Gamma(n/2+1)} \ge \text{Vol}(K) \implies D(K) \ge \frac{2}{\sqrt{\pi}} [\Gamma(n/2+1) \text{Vol}(K)]^{1/n}. $$ Using $\Gamma(n/2+1)\ge\sqrt{\pi n}(n/(2e))^{n/2}$ we can obtain $$ \frac{D(K)}{\text{Vol}(K)^{1/n}} \ge \frac{2 \sqrt{\pi n}^{1/n} }{\sqrt\pi}\sqrt\frac{n}{2e} \ge \sqrt\frac{2n}{\pi e} = C\sqrt n. $$ This bounds represents the extremal case, $m\to\infty$ but is not very useful for small $m$.
Now consider $K_s$ to be a regular $n$-simplex with $m=n+1$ and side-length $D(K_s^n)$. Then using the simplex volume formula and $n!\ge \sqrt{2\pi n}(n/e)^n$, $$ \frac{D(K_s^n)}{\text{Vol}(K_s^n)^{1/n}} = \sqrt{2}\left(\frac{n!}{\sqrt{n+1}}\right)^{1/n} \ge \left(\sqrt\frac{2\pi n}{n+1}\right)^{1/n}\frac{n}{e} \ge \frac{n}{e} = C'n. $$ Question 1: Does this bound hold for all $K$ with $m=n+1$ (is the simplex extremal here)?
Question 2: Is there a similar result for intermediate $m$ with ratios somewhere between $\sqrt n$ and $n$?