I often hear that the regular simplex is "the least" symmetric convex body, and I've heard that there are some measures of symmetry of a body, that the simplex minimizes.
Could you please explain or refer me to what methods / measurements there are that measure a convex body's symmetry?


I'll give some context - I have some function on convex bodies and I want to run a computer simulation to find which polytopes with volume 1 and k vertices minimize this functional.
It is conjectured that of all bodies of volume 1 a ball will minimize, and I want to gather evidence for or against this conjecture.
I'll have to measure symmetry and/or maybe, how close to ellipsoid-like shape a convex polytope is.


Thanks



Measurement of "symmetry" of a convex body
The type of symmetry for which the simplex (not necessarily regular) is usually called "the least symmetric convex body" is the symmetry of reflection about a point (e.g. $x\mapsto-x$). There are a few measures of this symmetry that the simplex minimizes. For any convex body $K\subset\mathbf{R}^n$, consider the following symmetric bodies:


$$ D = \tfrac12 (K-K) $$


$$ C = \mathrm{conv}\{K\times \{0\}, -K\times\{1\}\}\subset\mathbf{R}^{n+1} $$


$$ R\_a = \mathrm{conv}\{K, 2a-K\} $$


and let $R$ be the body of smallest volume among $\{R\_a\}$.


Note that if $K$ is symmetric under reflection about a point, then $|D|=|C|=|R|=|K|$. When $K$ is not symmetric, $|D|,|C|,|R|>|K|$. And, for a fixed volume of $|K|$, the simplex maximizes $|D|$, $|C|$, and $|R|$.


See [Convex Bodies Associated with a Given Convex Body](http://dx.doi.org/10.1112/jlms/s1-33.3.270), C. A. Rogers and G. C. Shephard (1957).


For your context it sounds like you want a measure of spherical symmetry. And since you want ellipsoids to be as "symmetric" as spheres, you probably want something affine invariant. I think you are looking for something like the Banach-Mazur distance to the sphere:


$$\delta(K,B) = \min\{t: B\subseteq TK+a\subseteq e^t B \text{ for some }T\in GL(n), a\in \mathbf{R}^n\}$$


The BM distance is usually defined for point-symmetric bodies, but there are versions out there without that assumptions (e.g. Macbeath AM (1951), "A compactness theorem for affine equivalence-classes of convex regions", Journal canadien de mathématiques. 3, 54–61).


