Bodies of minimal resistance under prescribed surface area

In Newton's problem of minimal resistance one seeks to minimize the functional [GRAPHICS] over a suitable class A of admissible functions. Here Omega subset of R-2 is the maximal cross section of a body travelling through a rarefied liquid. This variational problem can be derived from first principles in mechanics, see [10]. Various classes of admissible functions have been discussed Sor instance in [2, 5]. Since R is not coercive, one has to introduce some bound on the class of admissible functions, and one of the bounds that was suggested in [4, p. 259] Sor the case of radial functions was the surface area of the body. In this case EGGERS was able to conclude that a minimizer of R had to have conical shape. In the present paper we return to this optimal shape problem las well as to closely related questions) Sor a base domain Omega subset of R-n which is not necessarily a disk or ball. Throughout the paper Omega is assumed to be bounded and simply connected.