First-order magnitude conjecture for small convex bodies in hypermetric normed spaces

Let $E=(\mathbb{R}^n,\left\Vert\cdot\right\Vert)$ be a hypermetric normed space, and let $K\in\mathcal{K}^n$, the family of compact convex bodies in $\mathbb{R}^n$. Let $\mu_1^E(K)$ denote the first Holmes–Thompson intrinsic volume of $K$ in $E$. First-order magnitude conjecture.

$$\lim_{t\to0^+}\frac{\operatorname{Mag}\left(tK,\left\Vert\cdot\right\Vert\right)-1}{t}=\frac14\mu_1^E(K).$$

This asserts that the upper bounds for magnitude established earlier in the paper are sharp to first order as $t\to0^+$. It generalizes a conjecture for the Euclidean case; the limit is known when $E=\ell_2^n$, $n$ is odd, and $K$ has smooth boundary, while the conjecture is open in the other cases.