On the shadow boundary of a centrally symmetric convex body

We 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.

💡 Research Summary

The paper investigates the geometric and topological structure of two objects naturally associated with a centrally symmetric convex body $K\subset\mathbb R^{n}$ (equivalently, the unit ball of a Minkowski norm) and a fixed Euclidean direction $\mathbf x\in S^{n-1}$. The first object is the shadow boundary $\operatorname{Sh}_K(\mathbf x)$, defined as the set of points on the boundary of $K$ that attain the maximal support value in the direction $\mathbf x$: \