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:

  1. The neighbourhoods of interior points are just the euclidean neighbourhoods.
  2. 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?

  • 1
    $\begingroup$ Robert's answer says it all. It might be interesting to add that in complex analysis the ``cone-stump" used is called the Stolz angle. $\endgroup$ Sep 15 '16 at 17:44
  • $\begingroup$ Oh man. I did know that at some point. And now that I've googled Stolz angles again, I even remember having realised that this is exactly the same kind of convergence as the one I was looking at. Obviously I didn't write that insight down before and forgot it again... $\endgroup$ Sep 15 '16 at 18:11
  • $\begingroup$ Well, but two neighborhoods of a boundary point $x$ may intersect just by $\{x\}$, so $\{x\}$ itself is a neighborhood --- am I right? $\endgroup$ Sep 16 '16 at 10:41
  • $\begingroup$ @IlyaBogdanov Yes, you're right. I should have been more careful with my definition. Robert and Margaret gave the answer to the question I should have been asking. I really wanted something like Stolz angles. $\endgroup$ Jan 14 '17 at 20:49

Non-tangential convergence. This is a common theme in e.g. boundary values of harmonic functions, Hardy-Littlewood maximal functions, etc.


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