  In discrete differential geometry, it is widely believed that the discrete
Gaussian curvature of a polyhedral vertex star equals the algebraic area of its
Gauss image. However, no complete proof has yet been described. We present an
elementary proof in which we compare, for a particular normal vector, its
winding numbers around the Gauss image and its antipode with its critical point
index. This index is closely related to the normal degree of the Gauss map. We
deduce how the number of inflection faces is related to the numbers of positive
and negative components in the Gauss image. The resulting formula significantly
limits the possible shapes of the Gauss image of a polyhedral vertex star. For
example, if a triangulated vertex star is in general position and its Gauss
image has no self-intersections, then it is either a convex spherical polygon
if the curvature is positive, or a spherical pseudo-quadrilateral if the
curvature is negative.
