mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, Tarski's plank problem is a question about coverings of convex regions in ''n''-dimensional Euclidean space by "planks": regions between two hyperplanes. Tarski 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. A convex body K is called symmetric if it is centrally symmetric with respect to the origin; that is to say, a point x lies in ...

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

hyperplane In geometry, a hyperplane is a subspace whose dimension is one less than that of its ''ambient space''. For example, if a space is 3-dimensional then its hyperplanes are the 2-dimensional planes, while if the space is 2-dimensional, its hyper ...

''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 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 half-spaces bounded by the hyperplane, * S has at lea ...

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

infimum In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest low ...

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 polytope ...

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

table Table may refer to: * Table (furniture), a piece of furniture with a flat surface and one or more legs * Table (landform), a flat area of land * Table (information), a data arrangement with rows and columns * Table (database), how the table data ...

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

''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/ Geometry