We describe singularities of the convex hull of a generic compact smooth hypersurface in four-dimensional affine space up to diffeomorphism. It turns out that the boundary of the convex hull is the front of a Legendre variety. Its singularities are classified up to contact diffeomorphism.
@article{bwmeta1.element.bwnjournal-article-bcpv50z1p61bwm,
author = {Bogaevski, Ilia},
title = {Singularities of convex hulls as fronts of Legendre varieties},
journal = {Banach Center Publications},
volume = {50},
year = {1999},
pages = {61-74},
zbl = {0963.58016},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-bcpv50z1p61bwm}
}
Bogaevski, Ilia. Singularities of convex hulls as fronts of Legendre varieties. Banach Center Publications, Tome 50 (1999) pp. 61-74. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-bcpv50z1p61bwm/
[000] [1] I. A. Bogaevski, Singularities of convex hulls of three-dimensional hypersurfaces, Trudy Mat. Inst. Steklov. 221 (1998), 81-100 (in Russian); English transl.: Proc. Steklov Inst. Math. 221 (1998), 71-90.
[001] [2] A. B. Givental', Singular Lagrangian manifolds and their Lagrangian mappings, Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Ser. Sovremennye Problemy Matematiki 33, VINITI, Moscow, 1988, 55-112 (in Russian); English transl.: J. Soviet Math. 52 (1990), 3246-3278.
[002] [3] V. D. Sedykh, Functional moduli of singularities of convex hulls of manifolds of codimension 1 and 2, Mat. Sb. (N.S.) 119 (1982), 233-247 (in Russian); English transl.: Math. USSR-Sb. 47 (1984), 223-236. | Zbl 0509.58005
[003] [4] V. D. Sedykh, Singularities of convex hulls, Sibirsk. Mat. Zh. 24 (1983), no. 3, 158-175 (in Russian); English transl.: Siberian Math. J. 24 (1983), 447-461. | Zbl 0513.58008
[004] [5] V. D. Sedykh, Stabilization of the singularities of convex hulls, Mat. Sb. (N.S.) 135 (1988), 514-519 (in Russian); English transl.: Math. USSR-Sb. 63 (1989), 499-505.
[005] [6] V. D. Sedykh, The sewing of the swallowtail and the Whitney umbrella in a four-dimensional control system, Trudy GANG im. I. M. Gubkina, Oil and Gas, Moscow, 1997, 58-68 (in Russian).
[006] [7] V. M. Zakalyukin, Singularities of convex hulls of smooth manifolds, Funktsional. Anal. i Prilozhen. 11 (1977), no. 3, 76-77 (in Russian); English transl.: Functional Anal. Appl. 11 (1978), 225-227.
[007] [8] V. M. Zakalyukin, R. M. Roberts, Stability of Lagrangian manifolds with singularities (Russian), Funktsional. Anal. i Prilozhen. 26 (1992), no. 3, 28-34; English transl.: Funct. Anal. Appl. 26 (1992), 174-178.