Title:

Tight smoothing of the squared distance functions and applications to computeraided design and differential equations

We study the quadratic lower compensated convex transform C[l lambda]dist2(x, K) of the squared distance function to a nonempty, nonconvex closed set K ⊂ R[n]. These transforms, introduced in [20], provide C[1,1]smooth tightapproximations of the squared distance functions. We introduce a result to calculate explicit formulae for the lower transform of the squared distance functions. Our first main result is the geometric characterisation of critical points of the quadratic lower compensated convex transform Cllambda dist2(x,K), which shows that the geometric nonsmooth critical points of the squared distance function are identical to the usual critical points for the smoothed squared distance function. We then consider K ⊂ Rn be finite and classify the Morse indices of critical points of the lower transform C[l lambda]dist2(x, K) in R2 and R3, that is, classify critical points into nondegenerate critical points and degenerate critical points. This classification of Morse indices of critical points cannot be fully justified without knowing the behaviour of the lower transform of the squared distance function to finite sets. Therefore, we study some local properties of the lower transform C[l lambda]dist2(x, K) for the squared distance functions to finite sets to understand its behaviour in a small neighbourhood of critical points. Ae show that the lower transform C[l lambda]dist2(x, K) has a semiglobal property in that is, for lambda > 0 suffciently large, the lower transform of the squared distance function equals the squared distance function except on a small subset of a given bounded set. Under certain regularity assumptions of K, we establish that the lower transform of the squared distance function to finite sets has a semiglobal representation in C(K) under triangulations, which helps us to understand the role of the lower transform C[l lambda]dist2(x, K) in surface reconstruction.
