Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Vol. 50, No. 1, pp. 219233 (2009) 

On shadow boundaries of centrally symmetric convex bodiesÁ. G. HorváthBudapest University of Technology and Economics, H1521 Budapest, Hungary, email: ghorvath@math.bme.huAbstract: We discuss the concept of the socalled shadow boundary belonging to a given direction {\bf x} of Euclidean nspace $R^n$ lying in the boundary of a centrally symmetric convex body $K$. Actually, $K$ can be considered as the unit ball of a finite dimensional normed linear (= Minkowski) space. We introduce the notion of the general parameter spheres of $K$ corresponding to the above direction {\bf x} and prove that if all of the nondegenerate general parameter spheres are topological manifolds, then the shadow boundary itself becomes a topological manifold as well. Moreover, using the approximation theorem of celllike maps we obtain that all these parameter spheres are homeomorphic to the $(n2)$dimensional sphere $S^{(n2)}$. We also prove that the bisector (i.e., the equidistant set with respect to the norm) belonging to the direction ${\bf x}$ is homeomorphic to $R^{(n1)}$ iff all of the nondegenerate general parameter spheres are $(n2)$manifolds. This implies that if the bisector is a homeomorphic copy of $R^{(n1)}$, then the corresponding shadow boundary is a topological $(n2)$sphere. Keywords: bisector, general parameter sphere, shadow boundary, Minkowski spaces, celllike maps, manifolds Classification (MSC2000): 52A21, 52A10, 46C15 Full text of the article:
Electronic version published on: 29 Dec 2008. This page was last modified: 28 Jan 2013.
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