# Riemannian metric on a space of “not-quite-smooth” (hyper)surfaces?

Apologies if the title's a bit vague; I'll try to explain everything below, and please let me know if I should clarify anything.

I've been looking for a while at variational problems on polytopes. I'm particularly interested in the isoperimetric problem - maximizing volume relative to surface "area" under certain constraints, e.g. over polytopes of a fixed combinatorial type - and more generally, finding some sort of equivalent to surfaces of constant mean curvature in terms of the behaviour of the area functional.

My main goal has been to come up with a "nice" Riemannian metric on the space of polytopes, so that I can look at things like the gradient of area. Eventually I'd like to generalize objects like the shape operator in a way which relates to all this, so that the aforementioned gradient of area works out as being equivalent to mean curvature (I've asked some questions about this problem before, but all of them were vague and based on less understanding than I have currently, so I won't link to anything for now.)

I know that for a sufficiently smooth (hyper)surface in ℝn, since a normal exists at every point, variations of the surface can be thought of as normal vector fields; if we restrict ourselves to considering variations which preserve smoothness, then the vector field should also be smooth; and we can define an inner product on the space of variations by integrating the pointwise product of any two variations.

It feels like all this should extend to polytopes, but there's a problem in places where the surface bends sharply and no well-defined normal direction exists. The set of such points has measure zero so naively we might expect that we can ignore it when integrating; but I'm not so sure that we can, because we might have a variation where the instantaneous velocity of such points is infinite (even though it is finite for smooth points) and they therefore contribute a nonzero amount to the integral.

What I'd really like is some sort of general theory which encompasses both smooth surfaces and polytopes, as well as other surfaces "in between". It occurred to me to consider general surfaces as limits of smooth ones, but then how can you ensure that all the limits will be well-behaved? e.g. a sequence of smooth surfaces might converge to some other surface, but their normals might not converge at all! Or the normals might converge, but the inner product might not.

I also looked at the boundaries of convex bodies specifically. I know these to be fairly well-behaved, and it feels like it should be possible to extend the smooth theory to these surfaces at the very least. But even convex polytopes are not strictly convex, so they are arbitrarily close to non-convex bodies, so if we're looking at variations the non-convex theory still seems to be important.

My gut feeling is that there should be some space of surfaces which are not quite smooth (in general), but which are still sufficiently close to being smooth (as the boundary of a convex body always is) that it makes sense to define an inner product on variations in a way which generalizes the situation with smooth surfaces, and indeed to do other bits of differential geometry - define the shape operator, equate mean curvature with the gradient of the area functional, etc.

I'm sorely lacking in background knowledge, and also currently without university access to journals. Can anyone point me in the direction of some relevant literature or, better yet, help me gain a bit of insight into the theory I'm stumbling towards here?

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You can try first convex surfaces; parameterize these by their curvature, which you regard as a semi-positive measure on the surface; this covers both smooth an PL cases. The key is the existence-uniqueness theorem which holds in this context (Cauchy-Alexandrov-Pogorelov et al). Now, the space of such measures is a convex cone, so you have plenty of metrics on this cone, you would have to choose one most suitable for your purpose. –  Misha Feb 28 '13 at 3:39
Thanks, this sounds promising. I have heard of Pogorelov's result but have never seen it formally stated, and my knowledge of measure theory is almost nonexistent (I can barely remember an introductory course I took several years ago). Could you perhaps point me to somewhere where the curvature of a convex surface is clearly defined? For what it's worth, I'd like the metric to agree with the one I mentioned above (integral of pointwise product of normal vector fields) when restricted to smooth surfaces. –  Robin Saunders Feb 28 '13 at 4:36
Robin: You can start by looking at this: arxiv.org/pdf/1103.2383v1.pdf and the references they list (I would stick to the convex 2d case when things should be easier). –  Misha Feb 28 '13 at 5:54
Thanks, and sorry for not replying sooner. The reference to Federer was useful - his "sets of positive reach" capture some of what I'm looking for, though they seem to exclude things like the graph of |x| or of sin(x²). The concept of curvature measure is also nice, though it too seems too tied to convex notions and feels like it could be generalized further. Plenty of food for thought, though, so thanks again! –  Robin Saunders Mar 15 '13 at 0:43