‘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.

About Dmitry Ostrovsky

PhD student with interests in statistics, optimization, and machine learning.
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