I just posted this question as a comment to the question [Hypersurfaces and Elliptic Points](https://mathoverflow.net/questions/52554/hypersurfaces-and-elliptic-points)
but I don't know how many people will see it.


It's well known and easy to prove that a point on a closed hypersurface in $\mathbb{R}^n$ that has maximal distance from the origin has nonnegative second fundamental form.


However, this is apparently not necessarily true, if the hypersurface is only PL (piecewise linear). This raises the following question: What does the hypersurface look like near a point of maximal distance?



Point of maximal distance on a non-positively curved PL surface
It is easy to see that point of maximal distance has to be a vertex.
Further, all the edges starting from this vertex point in the directions of open half-space.


That is it — any vertex with the described property can appear as a point of locally maximal distance. (For global maximum, in addition it has to be a vertex of the convex hull of the surface.)


The angle around such vertex can be bidder than $2{\cdot}\pi$, but in this case the surface can not be locally convex at this vertex.


