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Journal of Convex Analysis 10 (2003), No. 1, 265--273
Copyright Heldermann Verlag 2003

Convex Bodies of Optimal Shape

G. Carlier
Université Paris IX Dauphine, Ceremade, Place de Lattre de Tassigny, 75775 Paris Cedex 16, France,

T. Lachand-Robert
Université Pierre et Marie Curie, Laboratoire d'Analyse Numérique, 75252 Paris Cedex 05, France,


Given a continuous function f : Sn-1 --> R, we consider the minimization of the functional
Integral over partial A of  f (nA) dHn-1
with respect to the subset A of Rn, included in a class of convex bodies defined by surface or shape conditions. This corresponds to non-parametric formulations of older problems, including Newton's problem of the body of minimal resistance, following an approach due to G. Buttazzo and P. Guasoni [J. Convex Analysis 4 (1997) 343--351]. We establish existence and uniqueness results and some characterizations of the minimizers.

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