Let $A\in\mathbb R^{n\times d}$ where $n \geq d$. Let $1\leq p < \infty$ and let $q$ be its Hölder conjugate exponent. Is it true that $$ \{x\in\mathbb R^d : \|Ax\|_p \leq 1\}^\circ = A^\top(B_q^n) $$ ? Here, $\circ$ denotes the polar and $B_p^n\subseteq\mathbb R^n$ is the unit $\ell_p$ ball.
It is known that the polar of the polytope $\{x\in\mathbb R^d : \|Ax\|_\infty \leq 1\}$ is given by the convex hull of $\pm$ the rows of $A$, i.e. $\{A^\top y : y\in\mathbb R^n, \|y\|_1\leq 1\} = A^\top(B_1^n)$ (see, e.g., Lemma 1.3 in these notes by Rothvoss). It is also simple to check that the polar of the ellipsoid $\{x\in\mathbb R^d : \|Ax\|_2 \leq 1\}$ is given by the ellipsoid $\{A^\top y : y\in\mathbb R^n, \|y\|_2\leq 1\} = A^\top(B_2^n)$.
The proof for the cases of $p\in\{1,2\}$ seem to rely on specific properties of those spaces, and I have found them difficult to generalize. It is easy to check one inclusion by Hölder's inequality; if $z = A^\top y$ for some $\|y\|_q \leq 1$, then for every $x$ with $\|Ax\|_p\leq 1$, $$\langle x,A^\top y\rangle = \langle Ax,y\rangle \leq \|Ax\|_p \|y\|_q \leq 1.$$
I have also poked around in the following papers, but they do not seem to give the result I am looking for as far as I can tell:
Meyer, Mathieu; Pajor, Alain, Sections of the unit ball of $\ell ^ n_ p$, J. Funct. Anal. 80, No. 1, 109-123 (1988). ZBL0667.46004.
Barthe, Franck; Naor, Assaf, Hyperplane projections of the unit ball of $\ell_{p}^{n}$, Discrete Comput. Geom. 27, No. 2, 215-226 (2002). ZBL0999.52003.
By the polarity theorem, it suffices to verify that $$ (A^\top(B_q^n))^\circ = \{x\in\mathbb R^d : \|Ax\|_p \leq 1\}. $$ This is significantly easier to verify. Indeed, if $x$ has $\|Ax\|_p \leq 1$, then Hölder's inequality implies that for all $\|y\|_q \leq 1$, $$ \langle x,A^\top y\rangle = \langle Ax,y\rangle \leq \|Ax\|_p\|y\|_q \leq 1 $$ so $\{x\in\mathbb R^d : \|Ax\|_p \leq 1\}\subseteq (A^\top(B_q^n))^\circ$, and if $x$ satisfies $$ \langle x,A^\top y\rangle = \langle Ax,y\rangle \leq 1 $$ for all $\|y\|_q\leq 1$, then it is in particular true for $y$ which witnesses the tightness of Hölder's inequality for $Ax$, so $\langle Ax,y\rangle = \|Ax\|_p \leq 1$.