In the geometry of convex polytopes, the Minkowski problem for polytopes concerns the specification of the shape of a polytope by the directions and measures of its facets.[1] The theorem that every polytope is uniquely determined up to translation by this information was proven by Hermann Minkowski; it has been called "Minkowski's theorem", although the same name has also been given to several unrelated results of Minkowski.[2] The Minkowski problem for polytopes should also be distinguished from the Minkowski problem, on specifying convex shapes by their curvature.
For any -dimensional polytope, one can specify its collection of facet directions and measures by a finite set of -dimensional nonzero vectors, one per facet, pointing perpendicularly outward from the facet, with length equal to the -dimensional measure of its facet.[3] To be a valid specification of a bounded polytope, these vectors must span the full -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, 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.[1]
Minkowski's uniqueness theoremEdit
It is a theorem of Hermann Minkowski 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.
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. It can be used to decompose arbitrary polytopes into simplices, and centrally symmetric polytopes into parallelotopes.[2]
GeneralizationsEdit
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.[4]
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.[4][5]
^ abKlain, Daniel A. (2004), "The Minkowski problem for polytopes", Advances in Mathematics, 185 (2): 270–288, doi:10.1016/j.aim.2003.07.001, MR 2060470
^ abAlexandrov, Victor (2004), "Minkowski-type and Alexandrov-type theorems for polyhedral herissons", Geometriae Dedicata, 107: 169–186, arXiv:math/0211286, doi:10.1007/s10711-004-4090-3, MR 2110761
^Alexandrov, A. D. (2005), Convex Polyhedra, Springer Monographs in Mathematics, Berlin: Springer-Verlag, ISBN3-540-23158-7, MR 2127379; 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
August 29, 2023
minkowski, problem, polytopes, geometry, convex, polytopes, concerns, specification, shape, polytope, directions, measures, facets, theorem, that, every, polytope, uniquely, determined, translation, this, information, proven, hermann, minkowski, been, called, . In the geometry of convex polytopes the Minkowski problem for polytopes concerns the specification of the shape of a polytope by the directions and measures of its facets 1 The theorem that every polytope is uniquely determined up to translation by this information was proven by Hermann Minkowski it has been called Minkowski s theorem although the same name has also been given to several unrelated results of Minkowski 2 The Minkowski problem for polytopes should also be distinguished from the Minkowski problem on specifying convex shapes by their curvature Contents 1 Specification and necessary conditions 2 Minkowski s uniqueness theorem 3 Blaschke sums 4 Generalizations 5 See also 6 ReferencesSpecification and necessary conditions EditFor any d displaystyle d dimensional polytope one can specify its collection of facet directions and measures by a finite set of d displaystyle d dimensional nonzero vectors one per facet pointing perpendicularly outward from the facet with length equal to the d 1 displaystyle d 1 dimensional measure of its facet 3 To be a valid specification of a bounded polytope these vectors must span the full d displaystyle 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 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 1 Minkowski s uniqueness theorem EditIt is a theorem of Hermann Minkowski 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 EditMain article Blaschke sum 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 It can be used to decompose arbitrary polytopes into simplices and centrally symmetric polytopes into parallelotopes 2 Generalizations EditWith 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 4 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 4 5 See also EditAlexandrov s uniqueness theorem Cauchy s theorem geometry References Edit a b Klain Daniel A 2004 The Minkowski problem for polytopes Advances in Mathematics 185 2 270 288 doi 10 1016 j aim 2003 07 001 MR 2060470 a b Grunbaum Branko 2003 15 3 Blaschke Addition Convex Polytopes Graduate Texts in Mathematics vol 221 2nd ed New York Springer Verlag p 331 337 doi 10 1007 978 1 4613 0019 9 ISBN 0 387 00424 6 MR 1976856 This description of how to specify the directions and measures follows Grunbaum 2003 Klain 2004 and Alexandrov 2004 uses slightly different information a b Alexandrov Victor 2004 Minkowski type and Alexandrov type theorems for polyhedral herissons Geometriae Dedicata 107 169 186 arXiv math 0211286 doi 10 1007 s10711 004 4090 3 MR 2110761 Alexandrov A D 2005 Convex Polyhedra Springer Monographs in Mathematics Berlin Springer Verlag ISBN 3 540 23158 7 MR 2127379 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 Retrieved from https en wikipedia org w index php title Minkowski problem for polytopes amp oldid 1027915175, wikipedia, wiki, book, books, library,
