A piece-wise constant or blocky signal can be defined as follows
Definition: Let $p,b\in\mathbb{N}$ such that $b\leq \left(p-1\right)$. Define the set of normalized blocky vectors as the following
\begin{equation} \mathcal{L}\left(p,b\right) = \left\{ \nu\in S^{p-1}: \sum\limits_{j=1}^{p-1} \mathbb{I}\left(\nu_j\ne \nu_{j+1}\right)\leq b\right\} \end{equation}
where $S^{p-1}$ denotes the unit sphere in $\mathbb{R}^p$ and
\begin{equation} \mathbb{I}\left(\nu_j\ne \nu_{j+1}\right) = \left\{\begin{array}{ll} 1 & \nu_{j+1} \ne \nu_j\\ 0 & Otherwise\\ \end{array}\right\} \end{equation}
What is the convex hull of $\mathcal{L}\left(p,b\right)$?
I think there exists a positive scalar $\zeta \leq 2$ such that the following set is not much larger than $conv\left( \mathcal{L}\left(p,b\right) \right)$, i.e. $ conv\left( \mathcal{L}\left(p,b\right) \right) \subseteq \mathcal{M}\left(p,b,\zeta\right) \subseteq \alpha conv\left( \mathcal{L}\left(p,b\right) \right)$ for some $\alpha > 1$.
\begin{equation} \mathcal{M}\left(p,b,\zeta\right) = \left\{ \nu\in S^{p-1}: \sum\limits_{j=1}^{p-1} \left\lvert\nu_{j+1}-\nu_j\right\lvert\leq \zeta\sqrt{b}\right\} \end{equation}
I proved that $conv\left(\;\mathcal{L}\left(p,b\right)\;\right)\subset\mathcal{M}\left(p,b,2\right)$.
Any idea to either prove my conjecture or find the form of convex hull.
