In this paper, we consider minimization of a real-valued function $f$ over $\bold R\sp {n+1}$ and
study the choice of the affine normal of the level set hypersurfaces of $f$ as a direction for minimization.
The affine normal vector arises in affine differential geometry when answering the question of what
hypersurfaces are invariant under unimodular affine transformations. It can be computed at points
of a hypersurface from local geometry or, in an alternate description, centers of gravity of slices. In
the case where $f$ is quadratic, the line passing through any chosen point parallel to its affine normal
will pass through the critical point of $f$. We study numerical techniques for calculating affine normal
directions of level set surfaces of convex $f$ for minimization algorithms.