# Is it possible to give an ordering of concave sets?

I don't know how to express the question in a "mathematically correct" way.

If I have objects (or sets) these can locally be concave, convex or flat.

Example:
a plate is less concave than a bowl which in turn is less concave than a glass etc...

In the mind this concept is quite intuitive but I have not been able to find a rule that formalizes it anywhere.

Is it possible to say that "one object is more concave than another"?

As a concept it reminds me a bit of the eccentricity of conics but I don't know if they are connected

My idea was to do a sort of integral average referring to the radius of the osculating circle, but I don't know if it makes sense

• A question for the "design phase" of this problem; how concave would you say that a saddle shape such as the graph of $z = x^2 -y^2$ is? What about the surface of a torus (which is convex in some places and concave in others)? Jan 28 at 23:40

The question is very open, so there are many ways to deal with it. Here are some possible paths to follow.

One ambiguity is whether "concave" is the right term. It is the complementary of convex, and it applies (when talking about 3D objects) to volumes, not surfaces.
So one may understand that the objects listed (plate, bowl, glass) are supposed to be filled. But if we fill a glass with water, the surface plus its filling make a convex object, not concave. And convexity is much more used than concavity anyway.

Another point is: the listed surfaces all have a boundary. But we may expect the order relation to work also for boundaryless surfaces, such as a sphere for example.

Another ambiguity is whether the object needs to be invariant by revolution around an axis, as all the objects you list. It actually simplifies the idea of filling the object, but we may also (like when defining what is a convex function from $$\mathbb R$$ to $$\mathbb R$$) say a surface is convex when its epigraph, i.e. the infinite part above it, in one direction, is convex. However this unboundedness does not fit well with the definition I intend to propose.

So let's restrict ourselves to objects that are either bounded convex volumes, or surfaces which can be closed by an upper plane lid and whose interior is then a bounded convex volume. All these objects have a volume measure $$v$$. (Obviously formal proofs should be included here, but I won't, because the question is too open to immediately go deep into one direction).

The object being bounded, it also has a height $$h$$ along $$z$$ axis. The convexity measure I propose, which can be used to define a total order relation, is then $$\frac {h^3} v$$. It is invariant by scaling.

One drawback of this definition is: it depends upon an axis. For objects that are bounded convex volumes (i.e. for which one does not need to put a lid to close the volume), the definition can avoid the choice of an axis by taking for $$h$$ the largest diameter of the object.