Non-optimality of conical parts for Newton's problem of minimal resistance in the class of convex bodies and the limiting case of infinite height

TL;DR

This paper investigates Newton's minimal resistance problem for convex bodies as height approaches infinity, revealing that conical boundary parts hinder optimality and challenging existing conjectures about optimal shapes.

Contribution

It demonstrates the non-optimality of conical boundary parts in the infinite height limit and disproves certain bodies previously conjectured to be optimal.

Findings

Conical boundary parts inhibit optimality in the infinite height limit.

Existence of solutions to the infinite height problem is established.

Certain conjectured optimal bodies are shown to be non-optimal.

Abstract

We consider Newton's problem of minimal resistance, in particular we address the problem arising in the limit if the height goes to infinity. We establish existence of solutions and lack radial symmetry of solutions. Moreover, we show that certain conical parts contained in the boundary of a convex body inhibit the optimality in the classical Newton's problem with finite height. This result is applied to certain bodies considered in the literature, which are conjectured to be optimal for the classical Newton's problem, and we show that they are not.