When thinking of some other problem, I stumbled upon the following innocently looking question that is natural enough to have been considered (and, possibly, solved) many years ago. However my attempts to search the literature for an answer resulted in next to nothing.

Let $K\subset\mathbb R^n$ be a convex cone and let $K^*=\{y:\langle x,y\rangle\ge 0\text{ for every }x\in K\}$ be its dual cone. Suppose that $K\supset K^*$ (or, if you prefer, even that $K=K^*$). What is the minimal possible ratio $\frac{|K\cap B|}{|B|}$ where $B$ is the unit ball in $\mathbb R^n$ when $n$ is large?

The answer should, probably, be of order $2^{-n}$ (positive orthant) but the best clean lower bound I can prove myself with my "homemade tools" is $(\sqrt 2+1)^{-n}$ (it can be improved a bit further to something like $2.317^{-n}$ but the argument gets somewhat messy and it is clear that this way won't lead to the optimal estimate).

Any help would be appreciated.

The probability for a symmetric matrix to be positive definite, mathoverflow.net/questions/118481/… $\endgroup$ – Denis Serre Apr 24 '18 at 18:32