1
$\begingroup$

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?

$\endgroup$
  • 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. $\endgroup$ – Margaret Friedland 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... $\endgroup$ – Johannes Hahn 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? $\endgroup$ – Ilya Bogdanov 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. $\endgroup$ – Johannes Hahn Jan 14 '17 at 20:49
4
$\begingroup$

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

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.