convex geometry In mathematics, convex geometry is the branch of geometry studying convex sets, mainly in Euclidean space. Convex sets occur naturally in many areas: computational geometry, convex analysis, discrete geometry, functional analysis, geometry of num ...

and the geometry of convex polytopes, the Blaschke sum of two polytopes is a polytope that has a

facet Facets () are flat faces on geometric shapes. The organization of naturally occurring facets was key to early developments in crystallography, since they reflect the underlying symmetry of the crystal structure. Gemstones commonly have facets cut ...

parallel to each facet of the two given polytopes, with the same measure. When both polytopes have parallel facets, the measure of the corresponding facet in the Blaschke sum is the sum of the measures from the two given polytopes. Blaschke sums exist and are unique up to

translation Translation is the communication of the meaning of a source-language text by means of an equivalent target-language text. The English language draws a terminological distinction (which does not exist in every language) between ''transla ...

, as can be proven using the theory of the Minkowski problem for polytopes. They can be used to decompose arbitrary polytopes into

simplices In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. ...

, 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. Although Blaschke sums of polytopes are used implicitly in the work of Hermann Minkowski, Blaschke sums are named for

Wilhelm Blaschke Wilhelm Johann Eugen Blaschke (13 September 1885 – 17 March 1962) was an Austrian mathematician working in the fields of differential and integral geometry. Education and career Blaschke was the son of mathematician Josef Blaschke, who taugh ...

, who defined a corresponding operation for smooth convex sets. The Blaschke sum operation can be extended to arbitrary convex bodies, generalizing both the polytope and smooth cases, using measures on the

Gauss map In differential geometry, the Gauss map (named after Carl F. Gauss) maps a surface in Euclidean space R3 to the unit sphere ''S''2. Namely, given a surface ''X'' lying in R3, the Gauss map is a continuous map ''N'': ''X'' → ''S''2 such that ' ...

Definition

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. As Hermann Minkowski proved, a finite set of nonzero vectors describes a polytope in this way if and only if it spans the whole

d

-dimensional space, no two are collinear with the same sign, and the sum of the set is the zero vector. The polytope described by this set has a unique shape, in the sense that any two polytopes described by the same set of vectors are translates of each other. The Blaschke sum

X\# Y

of two polytopes

X

and

Y

is defined by combining the vectors describing their facet directions and measures, in the obvious way: form the union of the two sets of vectors, except that when both sets contain vectors that are parallel and have the same sign, replace each such pair of parallel vectors by its sum. This operation preserves the necessary conditions for Minkowski's theorem on the existence of a polytope described by the resulting set of vectors, and this polytope is the Blaschke sum. The two polytopes need not have the same dimension as each other, as long as they are both defined in a common space of high enough dimension to contain both: lower-dimensional polytopes in a higher-dimensional space are defined in the same way by sets of vectors that span a lower-dimensional subspace of the higher-dimensional space, and these sets of vectors can be combined without regard to the dimensions of the spaces they span. For

convex polygon In geometry, a convex polygon is a polygon that is the boundary of a convex set. This means that the line segment between two points of the polygon is contained in the union of the interior and the boundary of the polygon. In particular, it is a ...

s and line segments in the Euclidean plane, their Blaschke sum coincides with their

Minkowski sum In geometry, the Minkowski sum (also known as dilation) of two sets of position vectors ''A'' and ''B'' in Euclidean space is formed by adding each vector in ''A'' to each vector in ''B'', i.e., the set : A + B = \. Analogously, the Minkowski ...

Decomposition

Blaschke sums can be used to decompose polytopes into simpler polytopes. In particular, every

d

-dimensional convex polytope with

n

facets can be represented as a Blaschke sum of at most

n-d

(not necessarily of the same dimension). Every

d

-dimensional

convex polytope can be represented as a Blaschke sum of parallelotopes. And every

d

-dimensional convex polytope can be represented as a Blaschke sum of

d

-dimensional convex polytopes, each having at most

2d

facets.

Generalizations

The Blaschke sum can be extended from polytopes to arbitrary bounded convex sets, by representing the amount of surface in each direction using a measure on the

of the set instead of using a finite set of vectors, and adding sets by adding their measures. If two bodies of constant brightness are combined in this way, the result is another body of constant brightness.

Kneser–Süss inequality

The volume

V(X\# Y)

of the Blaschke sum of two

d

-dimensional polytopes or convex bodies

X

and

Y

obeys an inequality known as the Kneser–Süss inequality, an analogue of the

Brunn–Minkowski theorem In mathematics, the Brunn–Minkowski theorem (or Brunn–Minkowski inequality) is an inequality relating the volumes (or more generally Lebesgue measures) of compact subsets of Euclidean space. The original version of the Brunn–Minkowski theor ...

on volumes of

s of convex bodies: :

V(X\# Y)^\ge V(X)^+V(Y)^.

References

{{reflist, refs= {{citation , last = Gronchi , first = Paolo , doi = 10.1007/s000130050224 , issue = 6 , journal = Archiv der Mathematik , mr = 1622002 , pages = 489–498 , title = Bodies of constant brightness , volume = 70 , year = 1998 {{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 , pages = 331–337 , publisher = Springer-Verlag , location = New York , series =

Graduate Texts in Mathematics Graduate Texts in Mathematics (GTM) (ISSN 0072-5285) is a series of graduate-level textbooks in mathematics published by Springer-Verlag. The books in this series, like the other Springer-Verlag mathematics series, are yellow books of a standard ...

, title = Convex Polytopes , title-link = Convex Polytopes , volume = 221 , year = 2003 {{harvtxt, Grünbaum, 2003, p. 339 {{citation , last = Schneider , first = Rolf , contribution = 8.2.2 Blaschke addition , contribution-url = https://books.google.com/books?id=kUaqCQAAQBAJ&pg=PA459 , doi = 10.1017/CBO9780511526282 , isbn = 0-521-35220-7 , mr = 1216521 , pages = 459–461 , publisher = Cambridge University Press, Cambridge , series = Encyclopedia of Mathematics and its Applications , title = Convex bodies: the Brunn-Minkowski theory , volume = 44 , year = 1993 Convex geometry Polytopes Binary operations