Frontiers Lectures

Lecture 3

Oriented distance function,
geometry,
and applications

October 13-17, 2003

Michel Delfour, FRSC

Centre de recherches mathématiques
and

Département de Mathématiques et de Statistique
Université de Montréal
Montréal, Canada


Abstract

According to C. Kuratovsky, the use of the distance function to define metrics on sets is due to the Romanian mathematician D. Pompéju in a paper in the Annales de Toulouse in 1905. The notion was popularized by F. Hausdorff in his book Grundzüge der Mengenlehre published in 1914 in Leipzig and is often referred to as the Hausdorff metric. This function provides an analytical description of a geometrical object, but is difficult to use in investigating the smoothness of the associated set. In their book in 1977 D. Gilbard and N.S. Trudinger provided the connection between the derivatives of the distance function to the boundary of a set and the curvatures of its boundary for sets of class C2. This was further formalized in 1981 by Krantz and Parks by using the signed distance function.

In 1994 M. Delfour and J.-P. Zolésio significantly sharpened and expanded the theory of distance functions and their relationship to the geometry of the underlying set. The notion of oriented distance function was introduced. This new terminology arises from the fact that, for smooth sets, the oriented distance function specifies the orientation of the exterior unit normal to the boundary and makes the curvature of the unit ball positive. The oriented distance function of a set coincides with its signed distance function, but it offers many technical advantages and avoids the frequent confusion with the properties of the distance function to a smooth submanifold.

New metrics and new compactness theorems were introduced to deal with sets ranging from sets with a non empty boundary to sets of class C. At the rough end of the spectrum, the new cracked sets were recently introduced for image processing problems. At the smooth end, the characterization of C1,1 sets by the C1,1 smoothness of the oriented distance function in a neighborhood of its boundary led to an intrinsic differential calculus on C1,1 hypersurfaces without local bases and Christoffel symbols and an equally intrinsic theory of Sobolev spaces on C1,1 hypersurfaces. This also resulted in significant breakthroughs and considerable simplifications in the theory of thin and asymptotic shells.

In this lecture we review the basic definitions and results on distance and oriented distance functions and introduce the associated differential calculus on an hypersurface. If time permits we shall look at some applications such as the Laplace-Beltrami operator.

References

M.C. Delfour, Tangential differential calculus and functional analysis on a C1,1- submanifold, in "Differential-geometric methods in the control of partial differential equations", R. Gulliver, W. Littman and R. Triggiani, eds., pp. 83--115, Contemp. Math, Vol. 268, AMS Publications, 2000.

M. C. Delfour and J.-P. Zolésio, Shape analysis via oriented distance functions, J. Funct. Anal. 123 (1994), no. 1, 129--201.

M. C. Delfour and J.-P. Zolésio, Differential equations for linear shells: comparison between intrinsic and classical models, in "Advances in the Mathematical Sciences - CRM's 25 years", Luc Vinet,ed., pp. 42-124, CRM Proc. Lecture Notes, vol. 11, Amer. Math. Soc., Providence, RI, 1997.

M. C. Delfour and J.-P. Zolésio, Shapes and Geometries: Analysis, Differential Calculus and Optimization, SIAM series on Advances in Design and Control, Society for Industrial and Applied Mathematics, Philadelphia, USA 2001.

D. Gilbarg and N.S. Trudinger, Elliptic partial differential equations of second order, Springer-Verlag, Berlin, New York, 1977 and 1983

S.G. Krantz and H.R. Parks, Distance to Ck hypersurfaces, J. Differential Equations 40 (1981), no. 1, 116-120.