Some applications of Weyl's Tube formula.

Dr Martin Lotz (The University of Manchester)

Frank Adams 1,

In 1840, Jacob Steiner showed that the volume of an r-neighbourhood of a 3-dimensional convex body could be written as a quadratic polynomial in r. A generalisation of this is a celebrated result by Hermann Weyl (1938), who showed that the volume of a small neighbourhood around a compact Riemannian submanifold is given by a polynomial whose degree is the dimension of the manifold. More importantly, he showed that the coefficients of this polynomial are independent of the embedding. Weyl’s tube formula became an important ingredient in Allendoerfer and Weil’s proof of the Gauss-Bonnet Theorem for hypersurfaces. In the last 30 years Weyl's tube formula has found applications in various fields, including numerical analysis and statistics. In this talk I will present an effective bound on the volume of tubular neighbourhoods of real and complex algebraic varieties, and discuss some applications.

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