This is sort of a definition question.
While a tvs is locally bounded if it contains a bounded neighborhood of the origin, a tvs is called locally convex if it contains a fundamental system of neighborhoods of the origin, say $\{U_n\}_{n\in\mathbb{N}}$, so that each $U_n$ is convex. Why the difference between the two? Is there an example of a tvs which has a convex neighborhood of the origin but not a fundamental system of neighborhoods of the origin? Or is a single convex neighborhood sufficient to generate an entire neighborhood basis?
I'm also curious as to why Rolewicz in Metric Linear Spaces defines a tvs to be locally $p$-convex if it has a fundamental system of the neighborhoods each of whose moduli of concavity are at most $2^{1/p}$. Is that not the same as having a neighborhood basis of $p$-convex sets?