On the topology of singularities of Maxwell sets

We determine new conditions for the coexistence of corank-one singularities of the Maxwell set of a generic family of smooth functions with respect to taking global minima (or maxima) in cases when this set does not have more complicated singularities. In particular, the Euler number of every odd-dimensional manifold of singularities of a given type is a linear combination of the Euler numbers of even-dimensional manifolds of singularities of higher codimensions. The coefficients of this combination are universal numbers (that is, they do not depend on the family and depend only on the classes of singularities).
We obtain these conditions as a corollary to the general coexistence conditions for corank 1 singularities of generic wave fronts which were found recently by the author. As an application, we obtain many-dimensional generalizations of the classical Bose formula relating the number of supporting curvature circles for a smooth closed convex generic plane curve to the number of supporting circles which are tangent to this curve at three points.