Swallowtail, Whitney umbrella and convex hulls (Q2070344)

The main result of the paper is the following: Consider a support hyperplane \(\pi\) of type \(A_1A_3\) to a smooth compact hypersurface \(M\) in \(\mathbb{R}^4\). Denote by \(P\) the type \(A_1\) endpoint of the support segment of \(\pi\). Let \(\Gamma\) be the boundary of the convex hull of the hypersurface \(M\). Then, any parallel projection of the germ at \(P\) of the singular point set of \(\Gamma\) to the hyperplane \(\pi\) for a generic \(M\) is diffeomorphic to the germ at zero of union of the cut swallowtail and the Whitney umbrella chamber (Theorem 1). This union is called by the author sailboat. Other author's paper directly connected to this topic are [\textit{V. D. Sedykh}, Funct. Anal. Appl. 11, 72--73 (1977; Zbl 0369.58009); Math. USSR, Sb. 47, 223--236 (1984; Zbl 0521.58008); Math. USSR, Sb. 63, No. 2, 499--505 (1989; Zbl 0679.58005); translation from Mat. Sb., Nov. Ser. 135(177), No. 4, 514--519 (1988); The sewing of the swallowtail and the Whitney umbrella in a four-dimensional control system. Moscow: Trudy GANG imeni I.M.Gubkina. 58--68 (1997); Izv. Math. 76, No. 2, 375--418 (2012; Zbl 1251.57021); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 76, No. 2, 171--214 (2012)].