This article introduces convex sets in finite-dimensional normed linear spaces equipped with a proximal relator. A proximal relator is a nonvoid family of proximity relations $\mathcal R_\delta$ (called a proximal relator) on a nonempty set. A normed linear space endowed with $\mathcal R_\delta$ is an extension of the Száz relator space. This leads to a basis for the study of the nearness of convex sets in proximal linear spaces.