What is the minimal volume of the intersection of a self-dual cone and the unit ball?

Question

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.

Answer 1

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I thought the following simple restatement of a bit narrower problem might help. Suppose that the convex cone $K$ is polyhedral: \begin{equation} K=\{x\in\R^n\colon a_i\cdot x\ge0\ \;\forall i\in\{1,\dots,N\}\}, \end{equation} where $a_1,\dots,a_N$ are nonzero vectors in $\R^n$ and $\cdot$ denotes the dot product (the case of a general convex cone $K$ can hopefully be done by approximation). Then, by Farkas' lemma (say), the dual cone $K^*$ is the conical span of the $a_i$'s. So, the condition $K\supseteq K^*$ means that $a_i\cdot a_j\ge0$ for all $i,j$, that is, all the angles between the vectors $a_i$ are $\le\pi/2$.

If now $N\le n$, then it should be comparatively easy to move the $a_i$'s so that they become pairwise orthogonal and in the process the cone $K^*$ (which is the conical span of the $a_i$'s) only increases, so that $K$ only decreases, and then so does its "spherical angle" measure $\mu(K):=\frac{|K\cap B|}{|B|}$. Thus, in the case when $N\le n$, we will have $\mu(K)\ge1/2^n$, as desired.

Then it will "only" remain to consider the case when $N>n$. Then one may try to find movements of $a_i$'s which will bring all of them into the conical hull of $n$ of the $a_i$'s without increasing $\mu(K)$; this certainly seems much more difficult to do than the above. Another thing to try here may be to embed the $a_i$'s into, or approximate them by vectors in, or otherwise map them into, $\R^N$.