Fractional Planks

نویسندگان

  • Ron Aharoni
  • Ron Holzman
  • Michael Krivelevich
  • Roy Meshulam
چکیده

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 <n 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 ∑ f j (xj ) ≥ 1 for all (x1, . . . , xn) ∈ S. The width of a function f j is defined as ∫ 1 0 f j (x) dx . In particular, we are interested in conditions on a convex subset of the unit cube in <n 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 <n we prove that the sum of widths must be at least 1/n (the true bound is conjectured to be 2/n).

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عنوان ژورنال:
  • Discrete & Computational Geometry

دوره 27  شماره 

صفحات  -

تاریخ انتشار 2002