I am working on the following problem set from Prof. Holger Dette: https://www.ruhr-uni-bochum.de/imperia/md/content/mathematik3/lehre/ss15/exercise6.pdf.
For $q \in (0, 1]$, define the $\ell_q$-ball as $${B}_q(R_q) = \left\{\theta \in \mathbb{R}^d\ \ \middle\vert\ \ \sum_{j = 1}^d|\theta_j|^q \leq R_q \right\},$$ and the weak $\ell_q$-ball with parameters $(C, \alpha)$ as $$B_{\alpha}^w(C) = \{\theta \in \mathbb{R}^d \mid |\theta|_{(j)} \leq Cj^{-\alpha}\ \text{for}\ j = 1, 2, \ldots, d\},$$where $|\theta|_{(1)} \geq |\theta|_{(2)}\geq \ldots \geq |\theta|_{(d)}$ are the ordered values $|\theta|_{(1)} , |\theta|_{(2)}, \ldots , |\theta|_{(d)}$.
(a) Show that the set ${B}_q(R_q)$ is star-shaped around the origin.
(b) For any $\alpha > 1/q$, show that there is a radius $R_q=R_q(C, \alpha)>0$ (independent of $d$) such that $B_{\alpha}^w(C) \subset {B}_q(R_q)$.
On the first part of the problem, I understand that a set $S$ is star-shaped around the origin if $\theta \in S \implies t\theta \in S$ for all $t \in [0, 1]$. Furthermore, my instinct is that the proof for the first statement should consist of some argument about the existence of some $\theta^*$ within and without ${B}_q(R_q)$, but despite having already spent quite some time thinking about this, I haven't obtained any coherent thoughts. Although I have some scattered thoughts and inclinations that I can't properly articulate, I really haven't gotten anywhere meaningful.
I would really appreciate any help towards a solution. Thanks.
As is typically the case, I have obtained something of a solution immediately after asking for one.
For the first part, we check that given $\theta = \langle \theta_1, \theta_2, \ldots, \theta_d \rangle \in {B}_q(R_q)$ and $t \in [0, 1]$, then we can write $$\sum_{i = 1}^d|t\theta_i|^q = t^q\sum_{i = 1}^d|\theta_i|^q \leq t^qR_q \leq R^q.$$Thus $t\theta \in {B}_q(R_q)$ if $\theta \in {B}_q(R_q)$, and therefore ${B}_q(R_q)$ is star-shaped.
For the second, we can upper bound the following sum thus $$\sum_{j = 1}^d|\theta_j|^q \leq \sum_{j = 1}^dCj^{-\alpha p} \leq C\sum_{j = 1}^{\infty}j^{-\alpha p},$$given some $d$ and $\theta \in B^{w}_{\alpha}$. It then suffices to select $R_q(C, \alpha) = C\sum_{j = 1}^{\infty}j^{-\alpha p}$, and we are done.