Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Clear visibility and the dimension of kernels of starshaped sets


Author: Marilyn Breen
Journal: Proc. Amer. Math. Soc. 85 (1982), 414-418
MSC: Primary 52A30; Secondary 52A35
DOI: https://doi.org/10.1090/S0002-9939-1982-0656115-5
MathSciNet review: 656115
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: This paper will use the concept of clearly visible to obtain a Krasnosel'skii-type theorem for the dimension of the kernel of a starshaped set, and the following result will be proved: For each $ k$ and $ n$, $ 1 \leqslant k \leqslant n$, let $ f(n,n) = n + 1$ and $ f(n,k) = 2n$ if $ 1 \leqslant k \leqslant n - 1$. Let $ S$ be a nonempty compact set in $ {R^n}$. Then for a $ k$ with $ 1 \leqslant k \leqslant n$, dim ker $ S \geqslant k$ if and only if every $ f(n,k)$ points of bdry $ S$ are clearly visible from a common $ k$-dimensional subset of $ S$. If $ k = 1$ or $ k = n$, the result is best possible. Moreover, if $ S$ is a compact, connected, nonconvex set in $ {R^2}$, then bdry $ S$ may be replaced by lnc $ S$ in the theorem.


References [Enhancements On Off] (What's this?)

  • [1] Marilyn Breen, A Krasnosel' skii-type theorem for points of local nonconvexity, Proc. Amer. Math. Soc. 85 (1982), 261-266. MR 652454 (83m:52008)
  • [2] -, $ K$-dimensional intersections of convex sets and convex kernels, Discrete Math. 36 (1981), 233-237. MR 675355 (84f:52006)
  • [3] -, The dimension of the kernel of a planar set, Pacific J. Math. 82 (1979), 15-21. MR 549829 (81h:52006)
  • [4] K. J. Falconer, The dimension of the kernel of a compact starshaped set, Bull. London Math. Soc. 9 (1977), 313-316. MR 0467536 (57:7392)
  • [5] Meir Katchalski, The dimension of intersections of convex sets, Israel J. Math. 10 (1971), 465-470. MR 0305237 (46:4367)
  • [6] M. A. Krasnosel'skii, Sur un critère pour qu'un domain soit étoilé, Mat. Sb. (61) 19 (1946), 309-310. MR 0020248 (8:525a)
  • [7] S. Nadler, Hyperspaces of sets, Dekker, New York, 1978. MR 0500811 (58:18330)
  • [8] Nick M. Stavrakas, The dimension of the convex kernel and points of local nonconvexity, Proc. Amer. Math. Soc. 34 (1972), 222-224. MR 0298549 (45:7601)
  • [9] F. A. Valentine, Convex sets, McGraw-Hill, New York, 1964. MR 0170264 (30:503)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 52A30, 52A35

Retrieve articles in all journals with MSC: 52A30, 52A35


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1982-0656115-5
Keywords: Starshaped sets, convex kernel
Article copyright: © Copyright 1982 American Mathematical Society

American Mathematical Society