Let X be a normed space. A set A ⊆ X is approximately convexif d(ta+(1-t)b,A)≤1 for all a,b ∈ A and t ∈ [0,1]. We prove that every n-dimensional normed space contains approximately convex sets A with and , where ℋ denotes the Hausdorff distance. These estimates are reasonably sharp. For every D>0, we construct worst possible approximately convex sets in C[0,1] such that ℋ(A,Co(A))=(A)=D. Several results pertaining to the Hyers-Ulam stability theorem are also proved.
@article{bwmeta1.element.bwnjournal-article-smv140i3p213bwm,
author = {S. Dilworth and Ralph Howard and James Roberts},
title = {On the size of approximately convex sets in normed spaces},
journal = {Studia Mathematica},
volume = {141},
year = {2000},
pages = {213-241},
zbl = {1039.26009},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-smv140i3p213bwm}
}
Dilworth, S.; Howard, Ralph; Roberts, James. On the size of approximately convex sets in normed spaces. Studia Mathematica, Tome 141 (2000) pp. 213-241. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-smv140i3p213bwm/
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