Quantifying convexity What methods exist to quantify convexity.  Yes, a set is convex if the the line between two points in the set is contained in the set, but is there a measure of how convex a set is?  If so, what is it?
 A: Indeed there are notions which say "how convex" (at least) the unit ball with respect to some norm is. First, there is the notion of uniform convexity. This basically says that the connecting line between two points on the boundary has to be strictly inside the body.
More refined notions can be gained from the modulus of convexity of the unit ball with respect to some norm. This basically measures how far inside the midpoint between two points on the boundary lies inside the unit ball depending on the distance of these two points. In formula: The modulus of convexity of a norm $\|\cdot\|$ is the function
$$
\delta(\epsilon) = \inf\{1 - \frac{\|x-y\|}{2}\ |\ \|x\|=\|y\| = 1,\ \|x-y\|\geq\epsilon\}.
$$
The slower $\delta$ tends to zero for $\epsilon\to 0$, the "more convex" the unit ball is.
As an example: For the unit ball for the 1-norm or the supremum norm, it holds $\delta(\epsilon) = 0$ for small $\epsilon$ and hence, these are not very convex things. If I remember correctly, for the two norm (or for any Hilbert space) it holds that $\delta(\epsilon) = 1 - \sqrt{1 - \epsilon^2/4}$ which is quadratic in $\epsilon = 0$, i.e. $\delta(\epsilon) = O(\epsilon^2)$. This is the slowest behavior possible.
I think that one could extend this notion to other convex sets with some effort but I don't know if this has been done.
Many books on Banach spaces and their geometry treat such topics.
A: You could in principle compare the set with its convex hull, which is a superset, and with its convex kernel, which is a subset. To quantify, you could compare area and volumes. I don't know any references for this approach, though.
Google found these papers, which may get your started:


*

*A measure of convexity for compact sets

*A Convexity Measurement for Polygons

*A new convexity measure for polygons
A: A set is either convex or not. However, one can try to measure how "symmetric" a convex set is. A classic paper about this is "Measures of Symmetry for Convex Sets" by Branko Grünbaum, pg. 233-270,in Convexity, Proceedings of Symposia in Pure Mathematics, Vol. VII, 1963, editor Victor Klee.
