The Krasnoselskii number
One of the best-known applications of Helly’s theorem (that is, actually, equivalent to Helly’s theorem, see [Bo]) is the following theorem, due to Krasnoselskii [K]:
Theorem 1 (Krasnoselskii): Let be an infinite compact set in . If every points of are visible from a common point, then is starshaped, that is there exists some that sees all points in .
Krasnoselskii’s theorem does not hold for non-compact sets. However, all known counter-examples satisfy the weaker requirement of being finitely starlike, namely that any finite subset of is visible from a common point. This led Peterson [P] to ask the following:
Does there exist a Krasnoselskii number such that for any infinite set , if every points of are visible from a single point, then is finitely starlike?
Marilyn Breen obtained positive results for special cases of sets that satisfy additional topological conditions [Br1,Br2,Br3], e.g., Krasnoselskii number 3 for closed sets in the plane. On the other hand, she proved that if exists, then it must be larger than .
Recently, Micha Perles and me (Chaya Keller) proved that under no restriction on the set, the Krasnoselskii number does not exist. More precisely, we proved that for any , there exists a set such that any points in are visible from a common point, but there exist points in that are not visible from a common point. But this counter example is absolutely not constructive (since it is based on a transfinite induction), and the resulting set admits no resonable topological structure.
This leads to the following two problems:
Problem 1: Can you show constructively that the Krasnoselskii number does not exist?
Problem 2: Can you find more natural topological conditions under which the Krasnoselskii number exists?
[Bo] Borwein, J., A proof of the equivalence of Helly’s and Krasnosselsky’s theorems,Canad. Math. Bull., 20 (1977), 35-37.
[Br1] M. Breen, Clear visibility, starshaped sets, and finitely starlike sets, J. of Geometry, 19 (1982), 183-196.
[Br2] M. Breen, Some Krasnoseleskii numbers for finitely starlike sets in the plane, J. of Geometry, 32 (1988), 1-12.
[Br3] M. Breen, Finitely starlike sets whose F-stars have positive measure, J. of Geometry, 35 (1989), pp.~19–25.
[K] M. A. Krasnoselskii, Sur un Critére pour Qu’un Domain Soit Étoilé, Math. Sb., 19 (1946), pp.309-310.
[P] B. Peterson, Is there a Krasnonselskii theorem for finitely starlike sets?, Convexity and Related Combinatorial Geometry, Marcel Dekker, New York, 1982, pp.81-84.