  In 2015, I. Soprunov and A. Zvavitch have shown how to use the
Bernstein-Khovanskii-Kushnirenko theorem to derive non-negativity of a certain
bilinear form $F_{\Delta}$, defined on (pairs of) convex bodies. Together with
C. Saroglou, they proved non-negativity of $F_K$ characterizes simplices, among
all polytopes. It is conjectured the characterization further holds among all
convex bodies. Towards this conjecture, several necessary conditions on $K$
(for non-negativity of $F_K$), were derived. We give a new necessary condition,
expressed with isoperimetric ratios, which provides a further step towards a
(conjectural) characterization of simplices among a certain subclass of convex
bodies.
