On shadow boundaries of centrally symmetric convex bodies

Á. G. Horváth

Abstract: We discuss the concept of the so-called shadow boundary belonging to a given direction {\bf x} of Euclidean n-space $R^n$ 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 non-degenerate general parameter spheres are topological manifolds, then the shadow boundary itself becomes a topological manifold as well. Moreover, using the approximation theorem of cell-like maps we obtain that all these parameter spheres are homeomorphic to the $(n-2)$-dimensional sphere $S^{(n-2)}$. 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 $R^{(n-1)}$ iff all of the non-degenerate general parameter spheres are $(n-2)$-manifolds. This implies that if the bisector is a homeomorphic copy of $R^{(n-1)}$, then the corresponding shadow boundary is a topological $(n-2)$-sphere.

Keywords: bisector, general parameter sphere, shadow boundary, Minkowski spaces, cell-like maps, manifolds

Classification (MSC2000): 52A21, 52A10, 46C15

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