Extensions of convex and semiconvex functions and intervally thin sets

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:365

Article

Tabor, Jacek¹  Tabor, Jozef² 

[1] Jagiellonian Univ, Inst Comp Sci, Lojasiewicza 6, PL-30348 Krakow, Poland

[2] Univ Rzeszow, Inst Math, PL-35310 Rzeszow, Poland

关键词: Semiconvexity;    Approximate convexity;    Extension;    Paraconvexity;   

DOI  :  10.1016/j.jmaa.2009.09.038

来源: Elsevier

【 摘 要 】

We call A subset of R-N intervally thin if for all x, y is an element of R-N and epsilon > 0 there exist x' is an element of B(x, epsilon), y' is an element of B(y, epsilon) such that [x', y'] boolean AND A = empty set. Closed intervally thin sets behave like sets with measure zero (for example such a set cannot disconnect an open connected set). Let us also mention that if the (N - 1)-dimensional Hausdorff measure of A is zero, then A is intervally thin. A function f is preconvex if it is convex on every convex subset of its domain. The consequence of our main theorem is the following: Let U be an open subset of R-N and let A be a closed intervally thin subset of U. Then every preconvex function f : U \ A -> R can be uniquely extended (with preservation of preconvexity) onto U. In fact we show that a more general version of this result holds for semiconvex functions. (c) 2009 Elsevier Inc. All rights reserved

【 授权许可】

Free