On the shadow boundary of a centrally symmetric convex body

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dc.creatorHorvath, Akos G.
dc.date2007-06-20
dc.date.accessioned2026-07-07T08:11:19Z
dc.date.available2026-07-07T08:11:19Z
dc.descriptionWe discuss the concept of the shadow boundary of a centrally symmetric convex ball $K$ (actually being the unit ball of a Minkowski normed space) with respect to a direction ${\bf x}$ of the Euclidean n-space $R^n$. We introduce the concept of general parameter spheres of $K$ corresponding to this direction and prove that the shadow boundary is a topological manifold if all of the non-degenerated general parameter spheres are, too. In this case, using the approximation theorem of cell-like maps we get that they are homeomorphic to the $(n-2)$-dimensional sphere $S^{(n-2)}$. We also prove that the bisector (equidistant set of the corresponding normed space) in the direction ${\bf x}$ is homeomorphic to $R^{(n-1)}$ iff all of the non-degenerated general parameter spheres are $(n-2)$-manifolds implying that if the bisector is a homeomorphic copy of $R^{(n-1)}$ then the corresponding shadow boundary is a topological $(n-2)$-sphere.
dc.description11 pages, 1 figures
dc.identifierhttps://arxiv.org/abs/0706.2958
dc.identifierhttp://arxiv.org/abs/0706.2958
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/132085
dc.subjectMetric Geometry
dc.subjectGeometric Topology
dc.subject57N16, 57N60
dc.titleOn the shadow boundary of a centrally symmetric convex body
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