Gram s relation for cone valuations

Transcription

1 Gram s relation for cone valuations Sebastian Manecke Goethe Universität Frankfurt joint work with Spencer Backman and Raman Sanyal July 31, 2018 Gram s relation for cone valuations Sebastian Manecke 1 / 17

2 Gram s relation for cone valuations A fundamental theorem of Euclidean geometry: β 2 γ 1 β 1 γ 5 γ 2 β 3 γ 3 γ 4 β4 β 5 Theorem For a polygon with n vertices, it holds n n n β i nπ + 2π = 0 β i γ i + 2π = 0 i=1 i=1 i=1 Gram s relation for cone valuations Sebastian Manecke 2 / 17

3 Cone angles A cone angle α assigns to each polyhedral cone C in R d a real number, such that Example α is a valuation (α(c C ) = α(c) + α(c ) α(c C ) whenever C C is convex), α is simple (α(c) = 0, whenever dim C < d), α is normalized (α(r d ) = 1). Take ν(c) := vol(c B d ) vol B d and this gives the usual angle. Gram s relation for cone valuations Sebastian Manecke 3 / 17

4 Main Theorem Theorem (1) Let P R d be a polytope and α be a cone angle. For i = 0,..., d 1, let α i (P) := α(â(f, P)), F P, dim F =i where Â(F, P) is the interior cone at F. Then the only linear relation among these numbers is the Gram-relation: d 1 ( 1) i α i (P) = ( 1) d+1. i=0 Gram s relation for cone valuations Sebastian Manecke 4 / 17

5 Interior angles Let F be a face of a polytope P. Define the interior cone Â(F, P) by the following procedure: Â(F, P) = cone(p q) + cone(p q). Move the origin to a point in the relative interior of F. Take the conical hull of P. (Minkowski-) add the dual of the affine hull (to make the cone full dimensional if P is not full-dimensional). F o o o P Gram s relation for cone valuations Sebastian Manecke 5 / 17

6 C 1 D 1 D 2 C 2 P D 5 D 4 C 5 C 3 D 3 C 4 Example α 0 (P) = α(c 1 ) + α(c 2 ) + α(c 3 ) + α(c 4 ) + α(c 5 ) α 1 (P) = α(d 1 ) + α(d 2 ) + α(d 3 ) + α(d 4 ) + α(d 5 ) and α 0 (P) α 1 (P) = 1. If α = ν, we recover the original statement after multiplying with 2π. Gram s relation for cone valuations Sebastian Manecke 6 / 17

7 Exterior angles Similarly, we define exterior cones q A(F, P) by taking the normal cone and adding the orthogonal complement: qa(f, P) := N F P + (N F P). P Gram s relation for cone valuations Sebastian Manecke 7 / 17

8 Exterior angles Theorem (1b) Let P R d be a polytope and α be a cone angle. For i = 0,..., d 1, let qα i (P) := α( A(F q, P)). F P, dim F =i Then the only linear relation among these numbers is: qα 0 (P) = 1. Gram s relation for cone valuations Sebastian Manecke 8 / 17

9 Partial proof of 1b P The normal cones of the vertices of a polytope form the normal fan. This is a complete fan. Thus: α 0 (P) = v vert(p) α( q A(v, P)) = v vert(p) α(n v P) = α(r d ) = 1. Gram s relation for cone valuations Sebastian Manecke 9 / 17

10 Zonotopes To prove, that there is only one linear relation, we need to find a set of polytopes, whose α i s (resp. qα i s) span the single hyperplane defined by the Gram-relation (resp. qα 0 (P) = 1). Definition A zonotope Z is the Minkowski-sum of line segments z 1,..., z n : Z = z z n R d. Example + + = z 1 z 2 z 3 Z Gram s relation for cone valuations Sebastian Manecke 10 / 17

11 Zonotopes II To every zonotope Z with face lattice F(Z) we can associate the lattice of non-empty faces F + (Z) := F(Z) \ { } and the lattice of flats L(Z): L(Z) := ({aff(f q) : F F + (Z), q F }, L 1 L 2 L 1 L 2 ) Example R 2 {x 1 = 0} {0} This gives a map L : F + (Z) L(Z), L(F ) = aff(f q) for some q F. Gram s relation for cone valuations Sebastian Manecke 11 / 17

12 Incidence Algebra The incidence algebra I(P) of a poset P = (P, ) is the vector space of all functions h : P P R, such that h(a, c) = 0 whenever a c and with multiplication (g h)(a, c) = g(a, b)h(b, c) for g, h I(P). Example a b c δ P (a, b) = 1 if a = b, δ P (a, b) = 0 otherwise, is the unit of the multiplication. ζ P (a, b) = 1 for a b. ζ P always has an inverse: µ P = ζ 1 P. Gram s relation for cone valuations Sebastian Manecke 12 / 17

13 Example For a cone angle α both α, qα are in I(F + (P)). Let F, F be two non-empty faces. Define α(f, F ) = α(â(f, F )) and qα(f, F ) = α( q A(F, F )). P qa(f, F ) F F Gram s relation for cone valuations Sebastian Manecke 13 / 17

14 Main Theorem II Theorem (2) For any cone angle α and any d-zonotope Z, we have α i (Z) = w co i (L(Z)), qα i (Z) = W co i (L(Z)), where wi co, wi co are the Co-Whitney numbers of first and second kind defined by: wi co = µ L(Z) (M, R d ), M L(Z) i Wi co = ζ L(Z) ({0}, M) = 1. M L(Z) i M L(Z) i Proof of Theorem 1. Find d zonotopes where you can calculate the Co-Whitney numbers easily (uniform matroids). Then do linear algebra to show that the Co-Whitney numbers span the single hyperplane defined by the Gram-relation (resp. qα 0 (P) = 1). Gram s relation for cone valuations Sebastian Manecke 14 / 17

15 How to prove theorem 2? Thus F L 1 (M) qα(f, Z) = 1 and summing over all M with rk M = i gives qα i (Z) = Wi co (L(Z)). Gram s relation for cone valuations Sebastian Manecke 15 / 17

16 How to prove theorem 2? You can prove the other half of Theorem 2 directly using the valuation property and the theory of hyperplane arrangements. It also follows from the fact, that L : F + (Z) L(Z) induces two adjoint maps L and L : F + (Z) L L(Z) I(F + (Z)) L I(L(Z)) I(F + (Z)) L I(L(Z)) You can then show that L ( α) = µ L(Z) and L (qα) = ζ L(Z) for any cone angle α. Gram s relation for cone valuations Sebastian Manecke 16 / 17

17 Thank you for your attention. Questions? Gram s relation for cone valuations Sebastian Manecke 17 / 17