## ‘Shear’ convex polyhedra: solution

Here I describe the solution to the problem from the last post. Spoilers!

As  $P_n$ one may take the simplex $0 \le x_1 \le\, ...\, \le x_n \le 1.$  Personally, I think about it as follows: start at the origin, then go to the point  $(1, 0,\, ...,\, 0),$  then to  $(1, 1, 0,\, ...,\, 0),$  etc. Finally take the convex hull of all the  $n+1$  vertices to obtain the simplex. The intuition is that each time forming a new vertex, we go out of the subspace spanned by all previous ones.

Obviously,  $W^*(P_n) = n.$  On the other hand, It is easy to obtain by some low-level convex optimization reasoning (it’s a good exercise though!) that  $w^*(P_n) = 2$  and in fact it is attained at any vertex.