Prove that there exists an absolutely continuous function $v(\cdot): I → R$ such that for any $y\in\mathbb{R}^n$ and $s,t\in [0,T]$, $$|dist (y, C(s)) − dist (y, C(t))| ≤ |v(s) − v(t)|$$ for all $s,t\in [0,T]$, where $C(t) := \{x\in \mathbb{R}^n|⟨a(t), x⟩ \leq b(t)\}$, $a(t)$ and $b(t)$ are two absolutely continuous functions on $[0,T]$.
I was thinking to split it into some cases as follows
$\bullet$ Case 1: If $y\in C(s)$ and $y\in C(t)$ then this case is obviously.
$\bullet$ Case 2: If $y\notin C(s)$ or $y\notin C(t)$ then we have to split it into three cases \begin{eqnarray}\label{d:1} \left\{\begin{matrix} \langle a(s),x\rangle >b(s)\\ \langle a(t),x\rangle \le b(t) \end{matrix}\right. \mbox{ or } \left\{\begin{matrix} \langle a(s),x\rangle \le b(s)\\ \langle a(t),x\rangle > b(t) \end{matrix}\right. \mbox{ or } \left\{\begin{matrix} \langle a(s),x\rangle > b(s)\\ \langle a(t),x\rangle > b(t) \end{matrix}\right. \end{eqnarray}