Consider a family of convex sets $\{K_n\}$ such that $K_n \subset \mathbb{R}^n$ for each $n$. The kinds of sets one might be considering could be, for instance,

$K_n$ is the cube of side $2A$, i.e., $K_n = [-A, A]^n$.

$K_n$ is the $n$ dimensional ball of radius $\sqrt{\lambda n}$.

Suppose we are interested in the rate of growth with respect to $n$, of the *volumes* of sets in the family $\{K_n\}$. We can define the parameter
$$v = \lim_{n \to \infty} \frac{1}{n}\log \text{Vol}(K_n),$$
which captures the exponential growth rate of volume. For the family of cubes, the parameter is $\log 2A$, and for the family of spheres it is $\frac{1}{2} \log 2 \pi e \lambda$.

Now instead of just volumes, suppose we are interested in the rate of exponential growth of intrinsic volumes of the family $\{K_n\}$. It seems intuitive to define the function $v:[0,1] \to \mathbb{R}$ as $$v(\theta) = \lim_{n \to \infty} \frac{1}{n} \log \mu_{n\theta}(K_n),$$ where $\mu_{n\theta}(K_n)$ is the $n\theta$-th intrinsic volume of $K_n$. Naturally $n\theta$ need not be an integer, but while taking the intrinsic volume we can round it off to the nearest integer. In case of the two examples considered,

- $v(\theta)$ for the family of cubes is: $H(\theta) + \theta \log 2A$,
- $v(\theta)$ for the family of spheres is: $H(\theta)+ \frac{\theta}{2} \log 2 \pi e \lambda + \frac{1- \theta}{2} \log (1 - \theta)$,

where $H(\theta) = -\theta \log \theta - (1 - \theta) \log (1 - \theta)$, is the binary entropy function. We can check that $v(1)$, which is the growth rate of volume, matches the $v$ defined earlier.

My questions are:

- What is this function $v(\theta)$? It seems to be a characteristic of the family $\{K_n\}$ considered as it tries to capture how this family is growing. Has this been encountered in the literature before?
- For what families of convex sets can one always guarantee the existence of $v(\theta)$?
- Is there any intuitive way to see why this limit function $v(\theta)$ should even exist for the two families considered above? (I explicitly used the formulae of intrinsic volumes for the above families to arrive at the corresponding $v(\theta)$).