This article is cited in 3 scientific papers (total in 3 papers)
On the topology of singularities of Maxwell sets
V. D. Sedykh
Gubkin Russian State University of Oil and Gas
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.
Key words and phrases:
Families of smooth functions, global minima and maxima, Maxwell sets, corank-one singularities of smooth functions, Euler number, convex curves, supporting hyperspheres.
MSC: Primary 58C05, 58K30; Secondary 53A04
Received: June 26, 2002
V. D. Sedykh, “On the topology of singularities of Maxwell sets”, Mosc. Math. J., 3:3 (2003), 1097–1112
Citation in format AMSBIB
\paper On the topology of singularities of Maxwell sets
\jour Mosc. Math.~J.
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Yomdin Y., “Generic singularities of surfaces”, Singularity Theory, 2007, 357–375
Haviv D., Yomdin Y., “Uniform approximation of near-singular surfaces”, Theoretical Computer Science, 392:1–3 (2008), 92–100
Houston K., van Manen M., “A Bose type formula for the internal medial axis of an embedded manifold”, Differential Geometry and Its Applications, 27:2 (2009), 320–328
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