The convex hull of an immersion

Romero Fuster, Maria del Carmen (1981) The convex hull of an immersion. University of Southampton, Doctoral Thesis.

Abstract

We consider an immersion of a compact orientable hypersurface in a Euclidean space. We take its convex hull 7((f) and define a panel structure on the boundary H(f)of X(f). This structure gives a decomposition of H(f) into panels that are related to the equilibrium positions of a generalised convex body of dimension m+1 and boundary H(f) when rolling or resting on any of its supporting hyperplanes. We study these structures in general. For n < 6 we find a residual subset of embeddings of M in R+l for which these structures satisfy some desirable conditions (proper panel structures or p-structures). On the other hand, we show that, in terms of convex hulls, any immersion can be reduced to a star-shaped embedding. This provides a large enough set of immersions satisfying those conditions and therefore defining p-structures on the boundaries of their convex hulls. Such p-structures also have the following property. Consider the Gauss map H(y) on the convex Cl hyperaphere H(f). Then the images of the panels by H(y) determines a Core stratification of the unit ire-sphere Sm. Some of the strata in this stratification are also determined by the images of those singularities of the Gauss map Y of fM lying on the exposed part of fM. We obtain a relation between the Euler characteristics of the strata of these stratifications. From this we deduce a formula relating the Euler characteristics of the panels and singular submanifolds of Y on H(f). This formula may be viewed in some cases as a generalization of the Euler formula for polyhedra. Finally, we discuss how to generalize these definitions and relations to the case of an immersion of a submanifold with highercodimension in any Euclidean space.

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