A geometric representation of the Morse fan
This article bases on the result by \textit{H. Th. Jongen} and \textit{D. Pallaschke} [Optimization 19, No. 3, 343--353 (1988; Zbl 0671.90099)] that around a nondegenerate critical point a piecewise smooth Morse function is topologically equivalent to the sum of a constant, a normalized nondegenerate quadratic form and the term \(\ell(y_1,...,y_k)\) where \(\ell\) is a continuous selection of the functions \(y_1,...,y_k\) and \(-\sum_{i=1}^k y_i\). The present paper studies a collection of cones in \(\mathbb R^k\), on which the latter continuous selections are linear. The construction of these cones goes back to \textit{S. G. Bartels, L. Kuntz} and \textit{S. Scholtes [Nonlinear Anal., Theory Methods Appl. 24, No. 3, 385--407 (1995; Zbl 0846.49005)]. The considered collection of cones form a complete polyhedral fan and are here briefly called Morse fan. The main result of the article is that the Morse fan is a refinement of the normal fan of the polytope which results from taking the Minkowski sum of a pyramid \(P\) and of \(-P\), where \(P\) is the convex hull of the unit vectors \(e_1,\dots,e_k\) and their negative sum \(-\sum_{i=1}^k e_i\). Furthermore, several details about the structure of the Morse fan are mentioned.}
scientific article; zbMATH DE number 2168244