Constructive approximation of compact hypersurfaces

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Let M \subset \mathbb{R}^k be a compact manifold of codimension 1. We show that M can be well approximated by a part of an algebraic manifold.

Theorem 1.
For all \varepsilon > 0, there exists c > 0 and a polynomial function P defined on \mathbb{R}^k such that N := P^{-1}(0) \cap (-c,c)^k is diffeomorphic to M and

\sup_{x \in N} \inf_{y \in M} \|x - y\| < \varepsilon.

This was proved by Seifert in a 1936 german language paper. It was later generalized to the Nash-Tognoli theorem which implies that non-singular real algebraic sets have precisely the same topological invariants as compact manifolds [1].

Here, however, our motivations are more elementary and practical. Our proof of theorem 1 points towards constructive approximation processes and shows the separability of the space of all compact hypersurfaces, under an appropriate topology. This is relevant in statistics and computer graphics, for smooth hypersurface regression and reconstruction. Growing sequences of finite dimensional search spaces of smooth manifolds are required in these applications.

In preparation for our proof, in section 2, we also discuss the Jordan-Brouwer theorem, the orientability of M and that M = f^{-1}(0) for some function f having regular value 0. We show that these three facts are essentially equivalent, in the sense that any one can be quite easily obtained from another.

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