Let P be the convex hull of the three points (0, 0), (0, 1) and (k, 1 2 ) in R2, where k ∈ N. Show that P(2k−1) = PI but P(2k) = PI .  Let P ⊆ [0, 1]n be a polytope in the unit hypercube with PI = ∅....

Let P be the convex hull of the three points (0, 0), (0, 1) and (k, 1

2 ) in R2,

where k ∈ N. Show that P(2k−1) = PI but P(2k) = PI .

 Let P ⊆ [0, 1]n be a polytope in the unit hypercube with PI = ∅. Prove that

then P(n) = ∅.

Note: Eisenbrand and Schulz [2003] proved that P(n2(1+log n)) = PI for any

polytope P ⊆ [0, 1]n. See also Pokutta and Schulz [2010], and Rothvoß and

Sanità [2017].