I was looking through some old notes of mine and stumbled upon a question I had wanted to ask a while back but never got around to it. Here it is now:
Consider a convex set $X\subseteq \mathbb{R}^n$ with non-empty interior and define a topology $\tau$ on $X$ as follows:
- The neighbourhoods of interior points are just the euclidean neighbourhoods.
- The neighbourhoods of a boundary point $x\in\partial X$ are generated by cone-"stumps", i.e. sets of the form $C\cap B_\epsilon(x)$ with $\epsilon>0$ where $C$ is a polyhedral cone pointing from the interior to its vertex $x$, i.e. $C = x+\sum_{l=1}^m \mathbb{R}_{\geq 0} (x_l-x)$ for some $x_l\in int(X)$. (In other words these cone-stumps lie completely in $int(X)$ except for $x$ itself if $\epsilon$ is small enough)
Why is this topology interesting to me? Well, while I was thinking about certain construction related to the Fourier-Laplace-transformation, I often found lemmas where I was considering some function defined on $X$ taking values somewhere nice (say holomorphic functions on some domain) that was continuous on $int(X)$, but the continuity condition $x_i \to x \implies f(x_i)\to f(x)$ for boundary points $x$ only held when I prohibited sequences approaching $x$ in a tangential way (and if I weakened the topology on the function space as well, but that's not relevant to my question). And that's exactly what this topology does: $x_i \xrightarrow{\tau} x$ holds iff $x_i\to x$ in the euclidean sense AND $x_i$ is eventually confined to some cone-stump.
Now my questions are simple:
Has this topology a name? Has it been observed before in the wild?
