mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, Tarski's plank problem is a question about coverings of convex regions in ''n''-dimensional

Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...

by "planks": regions between two

hyperplane In geometry, a hyperplane is a generalization of a two-dimensional plane in three-dimensional space to mathematical spaces of arbitrary dimension. Like a plane in space, a hyperplane is a flat hypersurface, a subspace whose dimension is ...

Alfred Tarski Alfred Tarski (; ; born Alfred Teitelbaum;School of Mathematics and Statistics, University of St Andrews ''School of Mathematics and Statistics, University of St Andrews''. January 14, 1901 – October 26, 1983) was a Polish-American logician ...

asked if the sum of the widths of the planks must be at least the minimum width of the convex region. The question was answered affirmatively by .

Statement

Given a

convex body In mathematics, a convex body in n-dimensional Euclidean space \R^n is a compact convex set with non- empty interior. Some authors do not require a non-empty interior, merely that the set is non-empty. A convex body K is called symmetric if it ...

''C'' in R^''n'' and a

''H'', the width of ''C'' parallel to ''H'', ''w''(''C'',''H''), is the distance between the two

supporting hyperplane In geometry, a supporting hyperplane of a Set (mathematics), set S in Euclidean space \mathbb R^n is a hyperplane that has both of the following two properties: * S is entirely contained in one of the two closed set, closed Half-space (geometry), h ...

s of ''C'' that are parallel to ''H''. The smallest such distance (i.e. the

infimum In mathematics, the infimum (abbreviated inf; : infima) of a subset S of a partially ordered set P is the greatest element in P that is less than or equal to each element of S, if such an element exists. If the infimum of S exists, it is unique ...

over all possible hyperplanes) is called the minimal width of ''C'', ''w''(''C''). The (closed) set of points ''P'' between two distinct, parallel hyperplanes in R^''n'' is called a plank, and the distance between the two hyperplanes is called the width of the plank, ''w''(''P''). Tarski conjectured that if a convex body ''C'' of minimal width ''w''(''C'') was covered by a collection of planks, then the sum of the widths of those planks must be at least ''w''(''C''). That is, if ''P''₁,…,''P''_''m'' are planks such that :

C\subseteq P_1\cup\ldots\cup P_m\subset \R^n,

then :

\sum_^m w(P_i)\geq w(C).

Bang proved this is indeed the case.

Nomenclature

The name of the problem, specifically for the sets of points between parallel hyperplanes, comes from the visualisation of the problem in R². Here, hyperplanes are just straight lines and so planks become the space between two parallel lines. Thus the planks can be thought of as (infinitely long) planks of wood, and the question becomes how many planks does one need to completely cover a

convex Convex or convexity may refer to: Science and technology * Convex lens, in optics Mathematics * Convex set, containing the whole line segment that joins points ** Convex polygon, a polygon which encloses a convex set of points ** Convex polytop ...

tabletop of minimal width ''w''? Bang's theorem shows that, for example, a circular

table Table may refer to: * Table (database), how the table data arrangement is used within the databases * Table (furniture), a piece of furniture with a flat surface and one or more legs * Table (information), a data arrangement with rows and column ...

diameter In geometry, a diameter of a circle is any straight line segment that passes through the centre of the circle and whose endpoints lie on the circle. It can also be defined as the longest Chord (geometry), chord of the circle. Both definitions a ...

''d'' feet can't be covered by fewer than ''d'' planks of wood of width one foot each.

References

* *{{citation, mr=0046672 , last=Bang, first= Thøger , title=A solution of the "plank problem" , journal=Proc. Amer. Math. Soc., volume= 2, year=1951, pages= 990–993 , doi=10.2307/2031721, issue=6, jstor=2031721 , url=http://www.ams.org/journals/proc/1951-002-06/S0002-9939-1951-0046672-4/, url-access=subscription Geometry