Let $n,d \ge 1$ be integers and $v_1,\dots, v_n \in \mathbb{R}^d$ be a set of vectors which spans $\mathbb{R}^d$. Let $\langle \cdot, \cdot \rangle$ be the standard dot product on $\mathbb{R}^d$.
Consider the convex program (P) below $$ \min_{x \in \mathbb{R}^d} \,\, \frac{1}{2}\langle x, x\rangle \quad \mbox{such that} \quad \langle v_i, x\rangle \ge 1, \, \forall i = 1,\dots, n. $$
Let $\hat{x}$ be the optimum solution of (P). By the KKT conditions of (P) and the (conical) Carathedory's theorem, there exists a subset $S \subseteq \{1,\dots, n\}$ where $|S| \le d$ and positive numbers $\alpha_s \in \mathbb{R}_{>0}$ indexed by $s \in S$ such that $$ \hat{x} = \sum_{s \in S} \alpha_s v_s. $$ My question: does there exists an $\tilde{x} \in \mathbb{R}^d$ such that $\alpha_s = \exp(\langle v_s, \tilde{x}\rangle)$ for each $s \in S$?
Background for anyone interested: I'm trying to figure out the existence part of Theorem 4 of https://arxiv.org/pdf/1710.10345.pdf