Let $z_1,\ldots,z_n$ be points in $\mathbb R^d$ and let $p \in [1,\infty]$. For any positive $c$ and $t$, consider the problems
\begin{align} &\min_{w \in \mathbb R^d} \|w\|^2_2 \text{ subject to }z_i^\top w \ge c\|w\|_p + 1\,\forall i \in [n],\tag{1}\\ &\min_{w \in \mathbb R^d} \|w\|_2 + t\|w\|_p \text{ subject to }z_i^\top w \ge 1\,\forall i \in [n].\tag{2} \end{align}
I read a document https://arxiv.org/pdf/1906.02931.pdf (see Remark 3.1 therein), and the author says that for any value of $c$, there exists a value of $t$ such that problems (1) and (2) are equivalent.
Question. How is that ?