Jean-Christophe Léger

• Doctor of Philosophy
• Professor at Lycée PG de Gennes-ENCPB

This paper provides (in french) a framework for an alternative demonstration of result of Khimshiashvili and Panina on the characterization of critical points of the area on the manifold of polygons with fixed sidelengths as being the cocyclical polygons. Other problems of the same class, with less constraints, are also alluded to and a numerical s...

We prove some monotonicity properties of the global Mumford–Shah minimizers as defined by Bonnet in [4]. The main consequence is that the only solutions for which the complement in the plane of the singular set is not connected correspond to lines and propellers. We also get a boundary version of the Mumford–Shah conjecture. Résumé Nous prouvons d...

We show that an analog of the age-Grayson-Hamilton Theorem for curves moving according to their mean curvature holds for the motion of quadrilaterals according to their Menger curvature.

For a Borel set E in R^n, the total Menger curvature of E, or c(E), is the integral over E^3 (with respect to 1-dimensional Hausdorff measure in each factor of E) of c(x,y,z)^2, where 1/c(x,y,z) is the radius of the circle passing through three points x, y, and z in E. Let H^1(X) denote the 1-dimensional Hausdorff measure of a set X. A Borel set E...

We extend A. Bonnet's results about the solutions of the global Mumford-Shah problem (see Bonnet (1996)) by replacing his connected assumption by more natural flatness assumptions.