Singularities of centre symmetry sets

Abstract

The center symmetry set (CSS) of a smooth hypersurface $S$ in an affine space $\mathbf{R}^n$ is the envelope of lines joining pairs of points where $S$ has parallel tangent hyperplanes. The idea stems from a definition of Janeczko, in an alternative version due to Giblin and Holtom. For $n = 2$ the envelope is always real, while for $n \ge 3$ the existence of a real envelope depends on the geometry of the hypersurface. In this paper we make a local study of the CSS, some results applying to $n \le 5$ and others to the cases $n = 2,3$. The method is to construct a generating function whose bifurcation set contains the CSS and possibly some other redundant components. Focal sets of smooth hypersurfaces are a special case of the construction, but the CSS is an affine and not a euclidean invariant. Besides the familiar local forms of focal sets there are other local forms corresponding to boundary singularities, and yet others which do not appear to have arisen elsewhere in a geometrical context. There are connections with Finsler geometry. This paper concentrates on the theory and the proof of the local normal forms for the CSS.