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 polygons and line segments in the Euclidean plane, their Blaschke sum coincides with their Minkowski sum. Decomposition Blaschke sums can be used to decompose polytopes into simpler polytopes. In particular, every -dimensional convex polytope with facets can be represented as a Blaschke sum of at most simplices (not necessarily of the same dimension). Every -dimensional centrally symmetric convex polytope can be represented as a Blaschke sum of parallelotopes. And every -dimensional convex polytope can be represented as a Blaschke sum of -dimensional convex polytopes, each having at most 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 Gauss map 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 of the Blaschke sum of two -dimensional polytopes or convex bodies and obeys an inequality known as the Kneser–Süss inequality, an analogue of the Brunn–Minkowski theorem on volumes of Minkowski sums of convex bodies: References Convex geometry Polytopes Binary operations