 ## Homework 41

### 1.Showthat the pyramid over a k-neighborly d-polytope is a k-neighborly (d+1)- polytope.  代写离散几何

2.(a) Show that a facet of a k-neighborly d-polytope is a k-neighborly (d 1)- polytope.

(b)Show that every n-neighborly 2n-polytope is simplicial.

3.Showthat a vertex-figure of a k-neighborly d-polytope is a (k  1)-neighborly (d  1)-polytope.

### 4.(a) A Petrie polygon of a convex 3-polytope P is a path along edges such that two successive edges, but not three, are edges of a 2-face of P .  代写离散几何

What are thelengths of the Petrie polygons of the five Platonic solids? (Here “length” means “combinatorial length”, that is, the number of edges in the path.)

(b)A Petrie polygon of a convex 4-polytope is a path along edges of P such that three successive edges, but not four, belong to a Petrie polygon of a facet of P . What are the lengths of the Petrie polygon of the 4-cube and 4-crosspolytope?

(c)Proposea good definition of “Petrie polygon” for any dimension dd 4 (say), and then find the length of the Petrie polygon of the d-cube.

5.(a) Show that each convex d-polytope P is flag-connected , in the sense that forany two distinct flags Φ and Ψ of P there is a sequence

Φ = Φ0, Φ1, . . . , Φk1, Φk = Ψ

of flags, such that Φj1 and Φj are adjacent (differ in exactly one face) for j = 1, . . . , k. (Remember Problem 5 of Homework Set 3.)  代写离散几何

Showthat each convex d-polytope P is strongly flag-connected , in the sense that for any two distinct flags Φ and Ψ of P there is a sequence

Φ = Φ0, Φ1, . . . , Φk1, Φk = Ψ

of flags, all containing Φ Ψ, such that Φj1 and Φj are adjacent (differ in exactly one face) for j = 1, . . . , k.

1Due Wednesday, March 27, 2019.