A note on bounded ratios

We prove that the set of bounded ratios $\text{BR}(X)$ on a semialgebraic set $X\subset {\mathbb{R}}_{>0}^n$ is the convex cone of linear forms that are nonnegative on the tropicalization $\text{trop}(X)$. In particular, it is a rational polyhedral convex cone. For $X$ the set of Lorentzian polynomials with fixed M-convex support, it is the dual to the set of M-convex functions. We record an explicit counterexample to a conjecture of Huang–Huh–Soskin–Wang on the bounded ratios on Lorentzian polynomials. The bounded ratio in the counterexample corresponds to the non-hypermetric clique-web facet ${\mathrm{C}\mathrm{W}}_7^1(1,1,1,1,1,-1,-1)$ of the cut cone on seven vertices.