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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.

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A New Excluding Condition Towards the Soprunov-Zvavitch Conjecture on Bézout-Type Inequalities

  • 会议论文

  • 发表日期：

    2026年

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  • Maud Szusterman

摘要

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.