Local properties of the surface measure of convex bodies

TL;DR

This paper investigates how local changes in the surface measure of convex bodies can produce local modifications of the bodies themselves, revealing a rich structure of perturbations and applying this to Newton's minimal resistance problem.

Contribution

It demonstrates the existence of extensive families of local surface measure perturbations that induce local convex body modifications under mild conditions.

Findings

Existence of line segment families of surface measure perturbations

Local perturbations lead to local convex body changes

Application to Newton's minimal resistance problem

Abstract

It is well known that any measure in S^2 satisfying certain simple conditions is the surface measure of a bounded convex body in R^3. 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.