Open problem session of HUJI-COMBSEM: Problem #2 Chaya Keller: The Krasnoselskii number

Chaya Keller

 

Marilyn Breen

This is our second post on the open problem session of the HUJI combinatorics seminar. The video of the session is here. Today’s problem was presented by Chaya Keller.

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 {S} be an infinite compact set in {\mathbb{R}^d}. If every {d+1} points of {S} are visible from a common point, then {S} is starshaped, that is there exists some {x_0 \in S} that sees all points in {S}.

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 {S} is visible from a common point. This led Peterson [P] to ask the following:

Does there exist a Krasnoselskii number {K(d)} such that for any infinite set {S \subset \mathbb{R}^d}, if every {K(d)} points of {S} are visible from a single point, then {S} 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 {K(2)} exists, then it must be larger than {8}.

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 {k \geq 2}, there exists a set {S \subset \mathbb{R}^2} such that any {2k+3} points in {S} are visible from a common point, but there exist {2k+4} points in {S} 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 {S} 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?

References

[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.

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4 Responses to Open problem session of HUJI-COMBSEM: Problem #2 Chaya Keller: The Krasnoselskii number

  1. Fedor Petrov says:

    “Micha Perles and me proved that under no restriction on the set, the Krasnoselskii number does not exist” – is there a text?

  2. Pingback: Open problem session of HUJI-COMBSEM: Problem #3, Ehud Friedgut – Independent sets and Lionel Levin’s infamous hat problem. | Combinatorics and more

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