Abstract

A direction dd is called a tangent direction to the unit sphere SS if the conditions s∈Ss\in S and aff⁡(s+d)\operatorname{aff}(s+d) is a tangent line to the sphere SS at ss imply that aff⁡(s+d)\operatorname{aff}(s+d) is a one-sided tangent to the sphere SS, i.e., it is the limit of secant lines at the point ss. A set MM is called convex with respect to a direction dd if [x,y]⊂M[x,y]\subset M whenever x,y∈Mx,y\in M, (y−x)∥d(y-x)\parallel d. It is shown that in an arbitrary normed space an arbitrary sun (in particular, a boundedly compact Chebyshev set) is convex with respect to any tangent direction of the unit sphere.

Author information

Contact details are reproduced from the original publication and may be historical.

Alexey R. Alimov

Faculty of Mechanics and Mathematics, Moscow State University
and: Steklov Math. Institute, Russian Academy of Sciences, Moscow, Russia

alexey.alimov-msu@yandex.ru

Evgeny V. Shchepin

Steklov Math. Institute, Russian Academy of Sciences, Moscow, Russia

scepin@mi.ras.ru

Keywords

  • Sun
  • Chebyshev set
  • directional convexity

Mathematics Subject Classification

41A65, 52A05

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Suggested citation

A. R. Alimov, E. V. Shchepin. “Convexity of Suns in Tangent Directions.” Journal of Convex Analysis 26 (2019), No. 4, 1071–1076. https://doi.org/10.68381/jca26058

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Published by Heldermann Verlag, 2019. Rights now held by Banach Press.