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The main purpose of this paper is to characterize the calibrability of bounded convex sets in IR N by the mean curvature of its boundary, extending the known analogous result in dimension 2. As a by-product of our analysis we prove that any bounded convex set C of class C 1,1 has a convex calibrable set K in its interior, and and for any volume V ∈ [|K|, |C|] the solution of the perimeter minimizing problem with fixed volume V in the class of sets contained in C is a convex set. As a consequence we describe the evolution of convex sets in IR N by the minimizing total variation flow.

Abstract. We prove existence and uniqueness of entropy solutions for the Cauchy problem for the quasilinear parabolic equation u t = div a(u, Du), where a(z, ξ ) = ∇ ξ f (z, ξ ), and f is a convex function of ξ with linear growth as ξ → ∞, satisfying other additional assumptions. In particular, this class includes a relativistic heat equation and a flux limited diffusion equation used in the theory of radiation hydrodynamics.

In this paper we compute the explicit evolution of the minimizing total variation flow when the initial condition is the characteristic function of a convex set in R 2 , or a finite number of them which are sufficiently separated. We also obtain some explicit solutions of the total variation formulation of the denoising problem in image processing. We illustrate these results with some experiments.

A novel approach for shape preserving contrast enhancement is presented in this paper. Contrast enhancement is achieved by means of a local histogram equalization algorithm which preserves the level-sets of the image. This basic property is violated by common local schemes, thereby introducing spurious objects and modifying the image information. The scheme is based on equalizing the histogram in all the connected components of the image, which are defined based both on the grey-values and spatial relations between pixels in the image, and following mathematical morphology, constitute the basic objects in the scene. We give examples for both grey-value and color images.