Journal of Convex Analysis 26 (2019), No. 4, 1373--1402
Copyright Heldermann Verlag 2019
Local Properties of the Surface Measure of Convex Bodies
Center for R&D in Mathematics and Applications, Dept. of Mathematics, University of Aveiro, Portugal
and: Institute for Information Transmission Problems, Moscow, Russia
It is well known that any measure in S2 satisfying certain simple conditions is the surface measure of a bounded convex body in R3. It is also known that a local perturbation of the surface measure may lead to a nonlocal perturbation of the corresponding convex body. We prove that, under mild conditions on the convex body, there are families of perturbations of its surface measure forming line segments, with the original measure at the midpoint, corresponding to local perturbations of the body. Moreover, there is, in a sense, a huge amount of such families. We apply this result to Newton's problem of minimal resistance for convex bodies.
Keywords: Convex sets, Blaschke addition, Newton's problems of minimal resistance.
MSC: 52A15, 52A40, 49Q10, 49Q20
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