Let X be a Banach space. Let C be a convex set in X. Then,
(1) A point $p\in C$ is an interior point if and only if for every $y\in X$, we have there exists some $\epsilon>0$ such that for every $t\in[-\epsilon,\epsilon]$, we have $p+ty\in C$
is this the same as the normal way of thinking about an interior point?
i.e. (2) there exists some $\delta>0$ such that $B_{||\,\cdotp\,||}(p,\delta)\subseteq C$.
And is the convexity of C required to make them the same definition if so? And how do we choose a $\delta$ if we want to show (1) imples (2)?
Thanks