Consider the following definition.

Let $C$ be a cycle of a simple graph $G$. We say that $C$ is *convex* if for any pair of distinct vertices $u,v \in V(C)$ $$ d_C(u,v) < d_{G-C}(u,v).$$

Is there any other name for such cycles? I was trying to find out some references/literature presenting results related to such cycles but I haven't found anything useful. I am mostly interested in the questions of whether such cycles have any other characterization and what is the structure of graphs that have many such cycles.