Fractional Planks

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 R ⁿ and touching all its sides is covered by planks. (A plank is a set of the form {(x ₁ , ..., x _n ): x _j ∈ 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 f ₁ , ..., f _n of nonnegative-valued measurable functions on [0,1] which cover the set S in the sense that ∑ f _j (x _j ) ≥ 1 for all (x ₁ , ..., x _n )∈ S . The width of a function f _j is defined as ∈t ₀ ¹ f _j (x) dx . In particular, we are interested in conditions on a convex subset of the unit cube in R 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 R ⁿ we prove that the sum of widths must be at least 1/n (the true bound is conjectured to be 2/n ).