# On the size of approximately convex sets in normed spaces

Studia Mathematica (2000)

• Volume: 140, Issue: 3, page 213-241
• ISSN: 0039-3223

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## Abstract

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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 $ℋ\left(A,Co\left(A\right)\right)\ge lo{g}_{2}n-1$ and $diam\left(A\right)\le C\surd n{\left(lnn\right)}^{2}$, 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.

## How to cite

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Dilworth, S., Howard, Ralph, and Roberts, James. "On the size of approximately convex sets in normed spaces." Studia Mathematica 140.3 (2000): 213-241. <http://eudml.org/doc/216765>.

@article{Dilworth2000,
abstract = {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 $ℋ(A,Co(A))≥log_2n-1$ and $diam(A)≤C√n(ln n)^2$, 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. },
author = {Dilworth, S., Howard, Ralph, Roberts, James},
journal = {Studia Mathematica},
keywords = {approximately convex set; approximately convex function; convex hull; diameter; Hausdorff distance; space of type },
language = {eng},
number = {3},
pages = {213-241},
title = {On the size of approximately convex sets in normed spaces},
url = {http://eudml.org/doc/216765},
volume = {140},
year = {2000},
}

TY - JOUR
AU - Dilworth, S.
AU - Howard, Ralph
AU - Roberts, James
TI - On the size of approximately convex sets in normed spaces
JO - Studia Mathematica
PY - 2000
VL - 140
IS - 3
SP - 213
EP - 241
AB - 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 $ℋ(A,Co(A))≥log_2n-1$ and $diam(A)≤C√n(ln n)^2$, 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.
LA - eng
KW - approximately convex set; approximately convex function; convex hull; diameter; Hausdorff distance; space of type
UR - http://eudml.org/doc/216765
ER -

## References

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