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• W2948302007 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 $intint_Omega (1 + |nabla u(x,y)|^2)^{-1} dx dy$ 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 $u_*$ solves the problem then $|nabla u_*(x,y)| ge 1$ at all regular points $(x,y)$ such that $u_*(x,y) > 0$. We prove that if the lower level set $L_0 = { (x,y): u_*(x,y) = 0 }$ has nonempty interior, then for almost all points of its boundary $(bar x, bar y) in partial L_0$ one has $lim_{stackrel{(x,y)to(bar x,bar y)}{u_*(x,y)>0}}|nabla u_*(x,y)| = 1$." @default.
• W2948302007 created "2019-06-14" @default.
• W2948302007 creator A5013883629 @default.
• W2948302007 date "2019-06-05" @default.
• W2948302007 modified "2023-09-27" @default.
• W2948302007 title "Behavior of convex surfaces near ridge points" @default.
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