The following affine invariant of convex bodies roughly measures how many of its linear images fit between the convex body and its dilates, if the dilations are taken with respect to the barycenter.

Given a convex body $K \subset \mathbb{R}^n$ with the barycenter at the origin and a real number $\lambda \geq 1$, define $$ \mathcal{R}(K,\lambda) := \{T \in GL(n;\mathbb{R}) : \frac{1}{\lambda} K \subset T(K) \subset \lambda K \}. $$ If $\mu$ is the Haar measure in $GL(n;\mathbb{R})$, set $\rho(K,\lambda) := \mu(\mathcal{R}(K,\lambda))$. This is the invariant.

Remark. The Haar measure is defined up to a multiplication by a constant, but I take as Haar measure the one whose value at a continuous, compactly supported function $f : GL(n;\mathbb{R}) \rightarrow \mathbb{R}$ is defined as $$ \int f(A) \frac{dA}{|\det(A)|^n} , $$ where $dA$ is the Euclidean measure on the space of $n \times n$ matrices (cf. Hewitt and Ross).

Basic properties of the invariant $\rho$.

1. $\rho(K, 1) = 0$ and $\rho(K, \lambda)$ is an increasing function of $\lambda$ tending to infinity as $\lambda$ tends to infinity.

2. If $S \in GL(n;\mathbb{R})$, $\rho(S(K),\lambda) = \rho(K,\lambda)$.

Indeed, $\mathcal{R}(S(K),\lambda) = S^{-1}\mathcal{R}(K,\lambda)S$ and $\mu$ is both right and left invariant (i.e., $GL(n,\mathbb{R})$ is unimodular).

3. If $K^*$ is the dual body of $K$, then $\rho(K^*,\lambda) = \rho(K,\lambda)$.

This follows from three observations: (1) duality is inclusion reversing; (2) $T(K)^* = T^{* -1}(K^*)$; (3) the Haar measure is invariant under inversion and transposition.

Questions.

1. Have you seen this invariant before? Is it a known invariant in a (possibly) new guise?

2. Can we derive other invariants by looking at the asymptotics (or derivatives) of $\rho(K,\lambda)$ as $\lambda \rightarrow 1$.

3. Intuitively, it one can fit many linear images of an ellipsoid $E$ in the shell bounded by $E/\lambda$ and $\lambda E$. This would suggest that ellipsoids maximize $\rho$. Can this be verified at least in dimension $2$?

4. Similarly, turning and deforming a simplex $\mathcal{T}$ seems to makes its vertices easily pop out of the shell bounded by $\mathcal{T}/\lambda$ and $\lambda \mathcal{T}$. This would suggest that simplices minimize $\rho$. Can this be verified at least in dimension $2$?