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We present a numerical procedure for solving the Minkowski problem, i.e., determining the convex set corresponding to a given curvature function. The method is based on Minkowski's isoperimetric inequality concerning convex and compact sets in R 3 . The support function of the target set is approximated in finite function space, so the problem becomes one of constrained optimization in R n , which in turn is solved by Newtonian (or other) iteration. We prove some properties of the optimization function and the constraining set and present some numerical examples.

A method has been developed for the determination of particle and crystal size distributions of rutile titanium dioxide pigments. This is based on a rigorous model for the turbidity (light extinction) spectra of rutile particles in dilute water suspension. The inversion of the rigorous model leads to an ill-posed problem and to a semi-rigorous method, in which the constraints of the solution have been determined empirically. The pigment particles are approximated as spheroids and a two-dimensional size distribution of particle width and ratio of length and width is calculated. Most of the particles are single crystals but a small amount is in aggregate form. A method to separate the distributions of aggregates and single crystals is proposed. The extinction cross section values for spheroids, needed in the model, are calculated by the T-matrix method. A comparison of the crystal size results with the values determined by transmission electron microscopy shows a very good accuracy for mean values of both the volume equivalent size and for the width, whereas the results for the means of the length/width ratio are much less accurate.