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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

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    $\begingroup$ I don't like to say that the simplex is the least symmetric body, but there are a lot of geometric invariants, where one extremal value is achieved by the ball and the other by the regular simplex. The simplest invariant that is conjectured to have this property is $V(K)V(K^*)$, where $K$ is a convex body and $K^*$ the polar body. $\endgroup$
    – Deane Yang
    Commented Jun 17, 2015 at 22:33
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    $\begingroup$ What is the functional you are trying to minimize? $\endgroup$
    – Moritz Firsching
    Commented Jun 18, 2015 at 4:40

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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, 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).

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