On a question by Firey concerning uniqueness

Abstract

We prove that the curvature equation ${\sum }_{j=1}^n{\alpha }_j{E}_j({\tau }_{\mathcal{M}})={\sum }_{j=1}^n{\alpha }_j{E}_j({\tau }_{\mathcal{N}})$ yields uniqueness up to translation for any two closed $C_{+}^2$ hypersurfaces $\mathcal{M},\mathcal{N}\hookrightarrow {\mathbb{R}}^{n+1}$ whenever $({\alpha }_1,\dots ,{\alpha }_n)\in {\mathbb{R}}_{\ge 0}^n\setminus \{0\}$ is log-concave and has no internal zeros. This gives an affirmative answer to a uniqueness question posed by Firey in the $C_{+}^2$ class.