Fractional planks

Ron Aharoni, Ron Holzman, Michael Krivelevich, Roy Meshulam

Research output: Contribution to journalArticlepeer-review

Abstract

In 1950 Bang proposed a conjecture which became known as "the plank conjecture": Suppose that a convex set S contained in the unit cube of ℜn and touching all its sides is covered by planks. (A plank is a set of the form {(x1, ..., xn): xj ∈ I} for some j ∈ {1, ..., n} and a measurable subset I of [0, 1]. Its width is defined as |I|.) Then the sum of the widths of the planks is at least 1. We consider a version of the conjecture in which the planks are fractional. Namely, we look at n-tuples f1, ..., fn of nonnegative-valued measurable functions on [0, 1] which cover the set S in the sense that Σfj(xj) ≥ 1 for all (x1, ...., xn) ∈ S. The width of a function fj is defined as ∫01fj(x)dx. In particular, we are interested in conditions on a convex subset of the unit cube in ℜn which ensure that it cannot be covered by fractional planks (functions) whose sum of widths (integrals) is less than 1. We prove that this (and, a fortiori, the plank conjecture) is true for sets which touch all edges incident with two antipodal points in the cube. For general convex bodies inscribed in the unit cube in ℜn we prove that the sum of widths must be at least 1/n (the true bound is conjectured to be 2/n).

Original languageEnglish
Pages (from-to)585-602
Number of pages18
JournalDiscrete and Computational Geometry
Volume27
Issue number4
DOIs
StatePublished - Jun 2002

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