# On the size of approximately convex sets in normed spaces

S. Dilworth; Ralph Howard; James Roberts

Studia Mathematica (2000)

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

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topDilworth, 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 -

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