Behavior of convex surfaces near ridge points

Abstract

The aim of this paper is twofold. First, we cut off a part of a convex surface by a plane near a ridge point and characterize the limiting behavior of the surface measure in $S^2$ induced by this part of surface when the plane approaches the point. Second, this characterization is applied to Newton's least resistance problem for convex bodies: minimize the functional $\int {\int }_{\Omega }(1+|\nabla u(x,y){|}^2{)}^{-1}dxdy$ in the class of convex functions $u:\Omega \to [0,M]$, where $\Omega \subset R^2$ is a convex body and $M>0$. It has been known that if $u_{\ast }$ solves the problem then $|\nabla u_{\ast }(x,y)|\ge 1$ at all regular points $(x,y)$ such that $u_{\ast }(x,y)>0$. We prove that if the lower level set $L_0=\{(x,y):u_{\ast }(x,y)=0\}$ has nonempty interior, then for almost all points of its boundary $(\bar{x},\bar{y})\in \partial L_0$ one has ${\lim }_{\stackrel{(x,y)\to (\bar{x},\bar{y})}{u_{\ast }(x,y)>0}}|\nabla u_{\ast }(x,y)|=1$.

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References in corpus (1)

• Newton's aerodynamic problem in media of chaotically moving particles

Cited by in corpus (1)

• Method of nose stretching in Newton's problem of minimal resistance