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The Convex Hull of a Hypersurface
Author(s) -
Robertson S. A.,
Romero-Fuster M. C.
Publication year - 1985
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms/s3-50.2.370
Subject(s) - mathematics , hypersurface , citation , regular polygon , combinatorics , mathematics education , library science , computer science , pure mathematics , geometry
m + 1 when rolling or resting on any of its supporting hyperplanes. These subsets of H(f) are called panels and the decomposition of H(f) so defined is called the panel structure. It turns out that there is no loss of generality in taking M = Sm and confining attention to smooth embeddings. This is because for any smooth immersion f:M-*Em + l there is a smooth embedding g: Sm -+ Em + 1 such that 3#>{g) = Jt?(f) and hence H(g) = H(f). We show that there is a residual subset of the space ${m) of smooth embeddings of Sm in Em + l on which the panel structure is 'well-behaved'. For such embeddings the panels are closely related to the strata of the core stratification discussed in (6). We also obtain a relation between the Euler numbers of the panels that may be regarded as a generalization of the Euler relation for polyhedra. The present version of this paper includes a number of improvements in both form and content suggested by the referee.