Magnitude's conjecture for intrinsic volumes of compact ℓ₁-convex sets

Let $A$ be a compact $\ell_1$-convex subspace of $\ell_1^N$, and let $V'_i(A)$ denote its $i$-th $\ell_1$-intrinsic volume. The $\ell_1$ magnitude–intrinsic-volume conjecture.

$$|A|=\sum_{i=0}^N2^{-i}V'_i(A).$$

The formula is established in the paper for cuboids, finite unions of suitable cuboids, and compact convex polygons in $\ell_1^2$, providing evidence for the conjecture. Its validity for all compact $\ell_1$-convex subspaces remains open.