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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).

Added: the question of which point-symmetric shape achieves the largest BM distance to the sphere is related to a major open problem in convex geometry, namely the asymptotics of the diameter of the BM space at large $n$. I don't know whether, for nonsymmetric shapes, it is known that the simplex achieves this. Perhaps someone more knowledgeable will comment.

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).

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).

Added: the question of which point-symmetric shape achieves the largest BM distance to the sphere is related to a major open problem in convex geometry, namely the asymptotics of the diameter of the BM space at large $n$. I don't know whether, for nonsymmetric shapes, it is known that the simplex achieves this. Perhaps someone more knowledgeable will comment.

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).

Added: the question of which point-symmetric shape achieves the largest BM distance to the sphere is related to a major open problem in convex geometry, namely the asymptotics of the diameter of the BM space at large $n$. I don't know whether, for nonsymmetric shapes, it is known that the simplex achieves this. Perhaps someone more knowledgeable will comment.

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).

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).

Added: the question of which point-symmetric shape achieves the largest BM distance to the sphere is related to a major open problem in convex geometry, namely the asymptotics of the diameter of the BM space at large $n$. I don't know whether, for nonsymmetric shapes, it is known that the simplex achieves this. Perhaps someone more knowledgeable will comment.