Compute lower bound on $\min_{E} \mathcal N(0,\sigma^2 I_n)(E)$ subject to $vol(E \cap H_n(r)) / vol(H_n(r)) \ge p$ where $H_n(r)$ is $n$-hemisphere

Question

Let $n \ge 2$ be an integer, which may be assumed to be very large. For $r > 0$, consider the hemi-sphere $H_n(r) := S_n(r) \cap (\mathbb R^+ \times \mathbb R^{n-1})$, where $$ S_n(r):= \{x \in \mathbb R^n \mid \|x\|_2 \le r\} $$ is the $n$-sphere of radius $r$. and consider the measure $\lambda_n$ define on borel-measurable subsets of $\mathbb R^n$ by $$ \lambda_n(E; r) := \frac{vol(E \cap H_n(r))}{vol(H_n(r))}. $$

Finally, let $\sigma > 0$ and $\mathcal N(0,\sigma^2 I_n)$ be the standard $m$-dimensional gaussian distribution with variance $\sigma^2$.

Question 1. For $p \in (0, 1)$, what is the borel-subset $\hat{E}$ of $\mathbb R^n$ which solves $$ \min_{E} \mathcal N(0,\sigma^2 I_n)(E)\text{ subject to } \lambda_n(E; r)\ge p. \tag{1} $$

Question 2. Same question with the hemisphere $H_n(r)$ replaced with the sphere $S_n(r)$.

Question 4. Same questions, but just provide the optimal objective value in (1), or a good lower-bound thereof.

Notes

• Of course, any helpful pointers / refs for addressing such problems are more than welcome

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Edit

• Question 3. Cancelled as it would make sense in general.

Answer 1

Let $\nu:=\mathcal N(0,\sigma^2 I_n)$ and $p\in[0,1]$.

Concerning Question 1: For the minimizing $E$, without loss of generality (wlog) we clearly have $E\subseteq H_n(r)$ and $\lambda_n(E;r)=p$. It is then clear that, subject to these conditions, wlog the set $E$ must consist of the points in $H_n(r)$ with the smallest density of $\nu$ wrt the Lebesgue measure; here, one might want to recall the Neyman–Pearson (NP) lemma. So, wlog $$E=E_*:=H_n(r)\setminus H_n(s), \tag{1} $$ where $s$ is the solution to the equation $\lambda_n(H_n(r)\setminus H_n(s);r)=p$, which can be rewritten as $$1-(s/r)^n=p. $$ (See details on (1) below.)

The answer to Question 2 is quite similar. This also answers Question 4.

As for Question 3, it does not quite make sense as stated, because the limits $\lambda_n(E;0)$ and $\lambda_n(E;\infty)$ will in general not exist, and also in view of the answers to Questions 1 and 2. Perhaps, Question 3 was intended as a relaxed, limit version of Questions 1 and 2.

Details on (1): Basically, here I will give a proof of the NP lemma, which I think is better than the one given in the linked Wikipedia article. Let $\lambda$ denote the Lebesgue measure on $\mathbb R^n$, and let $g:=\frac{d\nu}{d\lambda}$. Let real $c>0$ be such that for all $x\in H_n(r)$ we have $g(x)<c\iff x\in E_*$. Then for any Borel $E\subseteq H_n(r)$ $$(1_E-1_{E_*})(g-c)\ge0. \tag{2} $$ Assuming now that $\lambda(E)=\lambda(E_*)$, expanding the left-hand side of (2), and then integrating there wrt $\lambda$, we get $\nu(E)\ge\nu(E_*)$, as claimed.