Duality and the Descartes deficiency (Q748883)

Descartes has defined the angular deficiency \(\Delta\) of a convex rectilinear surface homeomorphic to a sphere \((\Delta =4\pi)\). A dual geometric invariant \(\Delta '\) for such a surface has been derived by \textit{B. Grünbaum} and \textit{G. Shephard} [Math. Gaz. 71, 214-216 (1987; Zbl 0631.52007)] (also \(\Delta '=4\pi)\). The present authors give a combinatorial interpretation of \(\Delta\) for any two-dimensional polyhedron and deduce an interpretation of \(\Delta '\) \((\Delta =\Delta '=2\pi \chi\), where \(\chi\) is the Euler characteristic of the polyhedron). For related results see the present authors [Enseign. Math., II. Ser. 27, 327-343 (1981; Zbl 0489.52009)].