Singularities of Centre Symmetry Sets

The center symmetry set (CSS) of a smooth hypersurface S inan affine space Rⁿ is the envelope of lines joining pairs ofpoints where S has parallel tangent hyperplanes. The idea stemsfrom a definition of Janeczko, in an alternative version dueto Giblin and Holtom. For n = 2 the envelope is always real,while for n > 3 the existence of a real envelope dependson the geometry of the hypersurface. In this paper we make alocal study of the CSS, some results applying to n 5 and othersto the cases n = 2,3. The method is to construct a generatingfunction whose bifurcation set contains the CSS and possiblysome other redundant components. Focal sets of smooth hypersurfacesare a special case of the construction, but the CSS is an affineand not a euclidean invariant. Besides the familiar local formsof focal sets there are other local forms corresponding to boundarysingularities, and yet others which do not appear to have arisenelsewhere in a geometrical context. There are connections withFinsler geometry. This paper concentrates on the theory andthe proof of the local normal forms for the CSS. 2000 MathematicsSubject Classification 57R45, 58K40, 32S25, 58B20.