The following invariants of "pointed" convex bodies (i.e., pairs consisting of a convex body and a distinguished point in its interior) roughly measure how many of its linear images fit between the convex body and its dilates, if the dilations are taken with respect to the distinguished point. Another possible viewpoint is that the invariant capture (or at least try to capture) the size of the set of affine transformations that fix the distinguished point and almost leave the body invariant.

Given a convex body $K \subset \mathbb{R}^n$ containing the origin (the distinguished point) in its interior and a real number $\lambda \geq 1$, define $$ \mathcal{R}((K,0),\lambda) := \{T \in GL(n;\mathbb{R}) : \frac{1}{\lambda} K \subset T(K) \subset \lambda K \}. $$

First invariant. If $\mu$ is the Haar measure in $GL(n;\mathbb{R})$, set $\rho((K,0),\lambda) := \mu(\mathcal{R}((K,0),\lambda))$.

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).

Second invariant. Let $GL^+(n;\mathbb{R})$ denote the group of linear transformations of $\mathbb{R}^n$ with positive determinant and define $$ \mathcal{R}^+((K,0),\lambda) := \{T \in GL^+(n;\mathbb{R}) : \frac{1}{\lambda} K \subset T(K) \subset \lambda K \}. $$ Set $\mathcal{D}(K,0)$ to be the infimum of all the values of $\lambda$ for which the inclusion $$ i : \mathcal{R}^+((K,0),\lambda) \longrightarrow GL^+(n;\mathbb{R}) $$ is a homotopy equivalence. This measures how much space is needed around $K$ to turn it at will.

Note that if we take the distinguished point (which we then move to the origin) to be the barycenter or the Santalo point of $K$, both of these invariant yield affine invariants.

Basic properties of the invariant $\rho$.

1.1 $\rho((K,0), 1) = 0$ and $\rho((K,0), \lambda)$ is an increasing function of $\lambda$ tending to infinity as $\lambda$ tends to infinity.

1.2 If $S \in GL(n;\mathbb{R})$, $\rho((S(K),0),\lambda) = \rho((K,0),\lambda)$.

Indeed, $\mathcal{R}((S(K),0), \lambda) = S^{-1}\mathcal{R}((K,0), \lambda)S$ and $\mu$ is both right and left invariant (i.e., $GL(n,\mathbb{R})$ is unimodular).

1.3 If $K^*$ is the dual body of $K$, then $\rho((K^*,0),\lambda) = \rho((K,0)\lambda)$.

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.

Basic properties of the invariant $\mathcal{D}$.

2.1 If $E$ is an ellipsoid centered at the origin, then $\mathcal{D}(E,0) = 1$.

This is because we may assume the ellipsoid is a ball and $GL^+(n;\mathbb{R})$ retracts by deformation onto $SO(n)$, which fixes the ball. Probably it is easy to see this characterizes ellipsoids.

2.2 If $S \in GL^+(n;\mathbb{R})$, then $\mathcal{D}(S(K),0) = \mathcal{D}(K,0)$.

As in 1.2, $\mathcal{R}^+((S(K),0), \lambda) = S^{-1}\mathcal{R}^+((K,0), \lambda)S$. Since $S$ has positive determinant and can be joined to the identity by a continuous curve of invertible transformations, the inclusions of $\mathcal{R}^+((K,0), \lambda)$ and $S^{-1}\mathcal{R}^+((K,0), \lambda)S$ into $GL^+(n;\mathbb{R})$ are homotopic.

2.3 $\mathcal{D}(K,0) = \mathcal{D}(K^*,0)$.

This is similar to (and easier than) 1.3.

Questions.

1. Have you seen these invariant before? Are they known invariants 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 $2$?

5. Are bodies easier to turn around about their barycenters than around any other point in their interior: Is $\mathcal{D}(K,\hbox{barycenter}) \leq \mathcal{D}(K,p)$ for any convex body $K$ and any point $p$ in the interior of $K$?

6. Are simplices harder to turn around than other bodies around their barycenters : Is $\mathcal{D}(\hbox{simplex},\hbox{barycenter}) \geq \mathcal{D}(K,\hbox{barycenter})$?