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.



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
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](https://en.wikipedia.org/wiki/Neyman%E2%80%93Pearson_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. 


