Definition talk:Minkowski Functional

Is there a concept of this which can be applied to a set which is specifically not convex and absorbing in a vector space? I am concerned that the context is unclear: I would recommend that we define the concept of a Minkowski functional with respect to a vector space first, then demonstrate that the concept cannot be applied to anything but such a set.

Note that I also have trouble with the fact that this is defined on a general vector space, while unless that space has at least a seminorm, it makes no sense if the scalar field of the vector space cannot be ordered (as happens when it is complex). --prime mover (talk) 07:39, 15 June 2023 (UTC)

Yeah I've sort of made a mess of this and I blame this to impatience on my part, since this is a necessary project to get Geometric Hahn-Banach in the weak topology. (which I need to prove the Goldstine theorem) I am not actually sure why I split this off from Definition:Minkowski Functional of Open Convex Set, since if ${\mathbf 0}_X$ is in the interior of a convex set, then that set is certainly absorbing. (say you have an open neighborhood $V$ of ${\mathbf 0}_X$ in a convex set $C$. Then for any $x \in X$ you have $x/n \to 0$, hence $x/N \in V$ for some large $N$, hence $x \in N V \subseteq N C$. I will put up a proof of this at some point) My understanding is that you can define the Minkowski functional on any subset, but convexity is opposed to make it sublinear ($\map {\mu_A} {x + y} \le \map {\mu_A} x + \map {\mu_A} y$) and the absorbing condition is to make sure that $\map {\mu_A} x < \infty$ for each $x \in X$. These are useful properties, but not wholly necessary. Rudin only deals with Minkowski functionals on convex and absorbing sets, however. And for my end goal of proving the Hahn-Banach Separation Theorem for locally convex spaces, (the proof is significantly different to the NVS case) all sets concerned will be convex. So I was just planning on getting through everything I need and coming back to this to generalise suitably.

With respect to the complex numbers not having an ordering, would you be more comfortable with writing $t \in \R_{> 0}$ in the infimum? Saying $t > 0$ for $t \in \C$ is more properly "$t \in \R$ and $t > 0$". Caliburn (talk) 07:57, 15 June 2023 (UTC)

If that is what is meant, then that is what is needed.

I would counsel against getting the fun stuff implemented and then going back to make the definitions rigorous, as this may result in lots of embarrassingly necessary rework after the fact. But in any case, at the moment none of this hangs together for a newbie in the field, rendering it more inaccessible than it needs to be. --prime mover (talk) 08:04, 15 June 2023 (UTC)

I don't think we risk a lack of rigour here. Really, the worst that could happen here is that results are not stated in full generality. Caliburn (talk) 08:40, 15 June 2023 (UTC)

No, the worst that can happen is that the reader doesn't have a clue what any of this area of mathematics is really about. This eventuality has already happened. :-) --prime mover (talk) 08:47, 15 June 2023 (UTC)