Harmonic Analysis and Convex Geometry, Fall 2013, October 4.

Instructor: Dmitry Ryabogin

Assignment 6.

1. Problem 1. Let Δ ⊂ ℝ^{3} be a tetrahedron. Prove that the following conditions are equivalent:

(i) all faces of Δ are congruent triangles,

(ii) all faces of Δ have equal perimeter,

(iii) all vertices of Δ have equal curvature,

(iv) the opposite edges of Δ have equal dihedral angles,

(v) all solid angles of Δ are equal.

Such Δ are called equihedral tetrahedra.

Hint: (ii) ⇒ (i) Write out equations for the edge lengths; (iii) ⇒ (i) From the Gauss-Bonnet theorem, conclude that the curvature of each vertex is equal to π. Now write out the equations for the angle sums around each vertex and inside each face; (iv) ⇒ (i) Use an argument based on the second proof of the Gauss-Bonnet Theorem given in class; (v) ⇒ (iv) Write out equations for the solid angles in terms of dihedral angles.

2. Problem 2. Let P ⊂ ℝ^{3} be a convex polytope. Denote by A and B the sums of all solid and dihedral angles, respectively. Prove that 2B − A = 2π(|F | − 2) where |F | is the number of faces in P.

Hint: Sum up over vertices.

3. Problem 3. Let P ⊂ ℝ^{3} be a convex polytope containing the origin in its interior. For a facet F of P, denote by α(F) the sum of the angles of F and by β(F) the sum of the angles of the projection of F onto a unit sphere centered at the origin. Finally, let ω(F) = β(F) − α(F). Prove that ∑_{F ⊂P} ω(F) = 4π.

4. Problem 4. Prove the theorem of Aleksandrov: Let P, Q ⊂ ℝ^{3} be two combinatorially equivalent convex polytopes with equal corresponding face angles. Then they have equal corresponding dihedral angles.

5. Problem 5*. Two convex polytopes P and Q in ℝ^{3} are called parallel if they are combinatorially equivalent and the corresponding faces are parallel. Prove the theorem of Aleksandrov: Assume that the perimeters or areas of parallel faces of parallel polytopes are equal. Then, there exists a ∈ ℝ^{3} such that P + a = Q.