On the size of approximately convex sets in normed spaces
| dc.creator | Dilworth, S. J. | |
| dc.creator | Howard, Ralph | |
| dc.creator | Roberts, James W. | |
| dc.date | 1999-08-17 | |
| dc.date.accessioned | 2026-07-07T05:30:21Z | |
| dc.date.available | 2026-07-07T05:30:21Z | |
| dc.description | Let X be a normed space. A subset A of X is approximately convex if $d(ta+(1-t)b,A) \le 1$ for all $a,b \in A$ and $t \in [0,1]$ where $d(x,A)$ is the distance of $x$ to $A$. Let $\Co(A)$ be the convex hull and $\diam(A)$ the diameter of $A$. We prove that every $n$-dimensional normed space contains approximately convex sets $A$ with $\mathcal{H}(A,\Co(A))\ge \log_2n-1$ and $\diam(A) \le C\sqrt n(\ln n)^2$, where $\mathcal{H}$ 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 $\mathcal{H}(A,\Co(A))=\diam(A)=D$. Several results pertaining to the Hyers-Ulam stability theorem are also proved. | |
| dc.description | 32 pages. See also http://www.math.sc.edu/~howard/ | |
| dc.identifier | https://arxiv.org/abs/math/9908086 | |
| dc.identifier | http://arxiv.org/abs/math/9908086 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/78966 | |
| dc.subject | Functional Analysis | |
| dc.subject | Metric Geometry | |
| dc.subject | 46B20(primary) 52A21 52A27 (secondary) | |
| dc.title | On the size of approximately convex sets in normed spaces | |
| dc.type | text |