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,

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

  2. $K_n$ 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,

  1. $v(\theta)$ for the family of cubes is: $H(\theta) + \theta \log 2A$,
  2. $v(\theta)$ 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:

  1. 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?
  2. For what families of convex sets can one always guarantee the existence of $v(\theta)$?
  3. 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 $v(\theta)$).

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.