Suppose one has a convex, bounded polytope P $\subset R^n$ and a strictly convex function $f$ defined everywhere on $R^n$. $f$ has a unique minimum; and suppose this minimum occurs somewhere strictly outside P, at a point x $\in R^n$.
Let's define two points on the boundary of P, which are in general distinct.
The first point, N, is the projection of x onto P, by which I mean that N is a point on P's boundary which minimizes the distance (in any norm, but say $l_2$) to x.
The second point, M, is the point on P's boundary which minimizes the function $f$ when restricted to P.
I think that N and M are always on the same facet of P. (By "on a facet" I include the (lower dimensional) "edges" of the facet.) My question is: is this true in general?
Here is an illustration of the following concrete example. Suppose we are in $R^2$, with $f(a,b) = a^2 + 4(b-4)^2$. Suppose that P is given by the four half-planes $a+b \le 7$, $2b-a \le 4$, $a \gt 0$, $b \gt 0$. This looks like:
Here's a picture of this situation:
The minimum of $f$ occurs at x = (0,4). In this case, we see that N and M are indeed on the same facet of P. Is this relationship true in general?