Consider a non empty bounded strictly convex body in n dimensions whose surface is a smooth manifold. We can relate points on the surfaces of two such bodies by requiring that their tangent planes be parallel. We take a unit ball as one of the bodies. Its surface comes with a measure and this 1-1 relation provides a measure for other such surfaces. This measure is of the Gaussian curvature of a subset of the surface. (Gaussian curvature)da is the same for equivalent surface subsets; equivalence defined by the relation and where da is an element of surface area. For n=2 the units are radians and for n=3 steradians.

We may extend the concept of a tangent plane to a non empty bounded convex body (allowing unsmooth bodies) by requiring that a tangent plane intersect the closure of the body but be disjoint from the interior of the body. If you don’t know point set topology, think “The plane touches but does not slice the body.”. In this extension the polyhedron has as tangents, planes thru a vertex, or planes containing an edge, or planes coincident with a face, as long as the plane does not slice the polyhedron into two pieces.

This relation between surface points and tangents is many-many; all the points of a face of a polyhedron share a tangent and all of the non slicing tangents thru a vertex share that surface point. The many points along an edge each have the same set of tangent planes. This provides a relation between surface points on two bodies; two points are related if they have parallel tangents. Nonetheless, this relation still transfers a surface measure between convex bodies. If one body is the unit ball the measure on the other body captures its total intrinsic curvature.

If our polyhedron is a cube then the imported measure ascribes π/2 to each of the eight corners and zero elsewhere; all of the curvature of the surface of a cube is concentrated in its corners. Tis the same with all simply connected polyhedrons. Non convex genus zero bodies could be allowed but then the relation between bodies would be oriented; something akin to a winding number would be needed. Some simply connected non-convex polyhedra will have corners with negative measure but the sum will always be 4π.

The set of rays beginning at a vertex of a polyhedron and extending into the polyhedron intersect a unit sphere about that vertex in a set, the content of whose supplement is the curvature at that vertex.

Here (now of here) is a pleasant introduction to these ideas. See too an obscure reference to 1943 Allendoerfer and Weil about some sort of n-dimensional extension.

### Gaussian curvature

If we are to carry this over to n-dimensions we must identify a more formal concept of Gaussian curvature. The boundary of n-dimensional convex bodies is embedded in n-space. The above definitions of tangent plane go over to n−1-dimensional subspaces that intersect the body’s closure but are disjoint from its interior. For smooth boundaries there is a quadratic form about the tangency point that gives the distance of the body from the tangent. This is sometimes called the second fundamental quadratic form. dξ = Hijduiduj where ui are parametric parameters describing the boundary.
All of this is striving for the n-dimensional version of the Gauss-Bonnet formula formula.