In the geometry of

convex polytope A convex polytope is a special case of a polytope, having the additional property that it is also a convex set contained in the n-dimensional Euclidean space \mathbb^n. Most texts. use the term "polytope" for a bounded convex polytope, and the wo ...

s, the Minkowski problem for polytopes concerns the specification of the shape of a polytope by the directions and measures of its facets. The theorem that every polytope is uniquely determined up to translation by this information was proven by

Hermann Minkowski Hermann Minkowski (; ; 22 June 1864 – 12 January 1909) was a German mathematician and professor at Königsberg, Zürich and Göttingen. He created and developed the geometry of numbers and used geometrical methods to solve problems in number t ...

; it has been called "Minkowski's theorem", although the same name has also been given to several unrelated results of Minkowski. The Minkowski problem for polytopes should also be distinguished from the Minkowski problem, on specifying convex shapes by their curvature.

Specification and necessary conditions

For any

d

-dimensional polytope, one can specify its collection of facet directions and measures by a finite set of

d

-dimensional nonzero vectors, one per facet, pointing perpendicularly outward from the facet, with length equal to the

(d-1)

-dimensional measure of its facet.This description of how to specify the directions and measures follows ; and uses slightly different information. To be a valid specification of a bounded polytope, these vectors must span the full

d

-dimensional space, and no two can be parallel with the same sign. Additionally, their sum must be zero; this requirement corresponds to the observation that, when the polytope is projected perpendicularly onto any

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 ...

, the projected measure of its top facets and its bottom facets must be equal, because the top facets project to the same set as the bottom facets.

Minkowski's uniqueness theorem

It is a theorem of

that these necessary conditions are sufficient: every finite set of vectors that spans the whole space, has no two parallel with the same sign, and sums to zero describes the facet directions and measures of a polytope. More, the shape of this polytope is uniquely determined by this information: every two polytopes that give rise to the same set of vectors are translations of each other.

Blaschke sums

The sets of vectors representing two polytopes can be added by taking the union of the two sets and, when the two sets contain parallel vectors with the same sign, replacing them by their sum. The resulting operation on polytope shapes is called the

Blaschke sum In convex geometry and the geometry of convex polytopes, the Blaschke sum of two polytopes is a polytope that has a facet parallel to each facet of the two given polytopes, with the same measure. When both polytopes have parallel facets, the meas ...

. It can be used to decompose arbitrary polytopes into simplices, and

centrally symmetric In geometry, a point reflection (point inversion, central inversion, or inversion through a point) is a type of isometry of Euclidean space. An object that is invariant under a point reflection is said to possess point symmetry; if it is invari ...

polytopes into parallelotopes.

Generalizations

With certain additional information (including separating the facet direction and size into a unit vector and a real number, which may be negative, providing an additional bit of information per facet) it is possible to generalize these existence and uniqueness results to certain classes of non-convex polyhedra. It is also possible to specify three-dimensional polyhedra uniquely by the direction and perimeter of their facets. Minkowski's theorem and the uniqueness of this specification by direction and perimeter have a common generalization: whenever two three-dimensional convex polyhedra have the property that their facets have the same directions and no facet of one polyhedron can be translated into a proper subset of the facet with the same direction of the other polyhedron, the two polyhedra must be translates of each other. However, this version of the theorem does not generalize to higher dimensions.

References

{{reflist, refs= {{citation , last = Alexandrov , first = A. D. , isbn = 3-540-23158-7 , mr = 2127379 , publisher = Springer-Verlag , location = Berlin , series = Springer Monographs in Mathematics , title = Convex Polyhedra , title-link = Convex Polyhedra (book) , year = 2005; see in particular Chapter 6, Conditions for Congruence of Polyhedra with Parallel Faces, pp. 271–310, and Chapter 7, Existence Theorems for Polyhedra with Prescribed Face Directions, pp. 311–348 {{citation , last = Grünbaum , first = Branko , authorlink = Branko Grünbaum , contribution = 15.3 Blaschke Addition , doi = 10.1007/978-1-4613-0019-9 , edition = 2nd , isbn = 0-387-00424-6 , mr = 1976856 , page = 331–337 , publisher = Springer-Verlag , location = New York , series = Graduate Texts in Mathematics , title = Convex Polytopes , title-link = Convex Polytopes , volume = 221 , year = 2003 {{citation , last = Alexandrov , first = Victor , doi = 10.1007/s10711-004-4090-3 , doi-access=free , journal = Geometriae Dedicata , mr = 2110761 , pages = 169–186 , title = Minkowski-type and Alexandrov-type theorems for polyhedral herissons , volume = 107 , year = 2004, arxiv = math/0211286 {{citation , last = Klain , first = Daniel A. , doi = 10.1016/j.aim.2003.07.001 , doi-access=free , issue = 2 , journal =

Advances in Mathematics ''Advances in Mathematics'' is a peer-reviewed scientific journal covering research on pure mathematics. It was established in 1961 by Gian-Carlo Rota. The journal publishes 18 issues each year, in three volumes. At the origin, the journal aimed ...

, mr = 2060470 , pages = 270–288 , title = The Minkowski problem for polytopes , volume = 185 , year = 2004 Polytopes Hermann Minkowski

Specification and necessary conditions

Minkowski's uniqueness theorem

Blaschke sums

Generalizations

See also

References