2018/05/28 by Alexander Plakhov, Plakhov, Alexander
Mathematics · #49Q10 #49Q20 #52A15 #52A40 #Analytic and geometric function theory #FOS: Mathematics #Geometric Analysis and Curvature Flows #Metric Geometry (math.MG) #Point processes and geometric inequalities
paper · pdf · doi:10.48550/arxiv.1805.11055
openalex publication_date 2018/05/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 a convex body, there are families of perturbations of its surface measure forming line segments, with the original measure at the midpoint, leading 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.