Open AccessSuns are convex in tangent directions
Author(s) -
А. Р. Алимов,
А. Р. Алимов,
E. V. Shchepin,
E. V. Shchepin
Publication year - 2019
Publication title -
doklady akademii nauk. rossijskaâ akademiâ nauk
Language(s) - English
Resource type - Journals
ISSN - 0869-5652
DOI - 10.31857/s0869-56524842131-133
Subject(s) - unit sphere , tangent , tangent vector , tangent space , mathematics , normed vector space , regular polygon , mathematical analysis , tangent cone , convex set , combinatorics , space (punctuation) , line (geometry) , tangent stiffness matrix , geometry , physics , convex optimization , computer science , operating system , stiffness matrix , finite element method , thermodynamics
A direction d is called a tangent direction to the unit sphere S of a normed linear space s S and lin(s + d) is a tangent line to the sphere S at s imply that lin(s + d) is a one-sided tangent to the sphere S, i. e., it is the limit of secant lines at s. A set M is called convex with respect to a direction d if [x, y] M whenever x, y in M, (y - x) || d. We show that in a normed linear space an arbitrary sun (in particular, a boundedly compact Chebyshev set) is convex with respect to any tangent direction of the unit sphere.