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Suppose I have a normed vectorspace $(X,\|.\|)$ and a (differential) path $\gamma:[0,1]\rightarrow X$. Can the Length of the curve be defined as $$L(\gamma)=\int_0^1\|\gamma'(t)\|\text{d}t$$ Or do other modifications have to be applied? In the wikipedia articles I found, all proofs were done via the euclidean norm, with no notion of arbitrary norms. The only thing I found that was more in general was the article about riemann submanifolds, but I am not happy with using something we have yet to learn.

CBenni
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The definition that you mention is fine, although I suspect you might want an affine vector space instead of a simple vector space. This just means you can have vectors based at any point and not just the origin. Of course it's fine with a vector space, but might seem a little unnatural.

If you are to progress your theory then you need to find an arc-length parameter, i.e. a parameter $s$ for which $||d\gamma/ds|| = 1$ for all $s$. After that, you will need to think about what curvature means.

You would be surprised about how varied the geometries can be. There is a geometry based on area instead of length. In this geometry you look for an arc-"length" parameter $s$ for which

$$ \det\left( \frac{d\gamma}{ds},\frac{d^2\gamma}{ds^2}\right) = 1$$

for all $s$. Notice that $\det$ measures the oriented area spanned by $d\gamma/ds$ and $d^2\gamma/ds^2$. It turns out that this arc-"length" parameter is given by:

$$s(t) = \int \det\left(\dot{\gamma},\ddot{\gamma}\right)^{1/3} \, dt \, . $$

Fly by Night
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    Thank you ;) I am basically just looking to calculate the length of a path in $(\mathbb{R}^2,\|.\|)$ where $\|.\|$ is any norm. I kept the question general in sake of meeting this pages requirements ;) – CBenni Dec 25 '12 at 15:49
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    @CBenni It was a very nice question, and I'm pleased to see you're thinking in general terms. If I were you then I'd chose a favourite norm and run with it. You'd be amazed by where it takes you. Like I said: arc-length is the first thing to generalise, and then curvature. After that you can talk about vertices and inflections. In the area case the vertices relate to unusually-high contact with ellipses and hyperbolae, while the inflections relate to contact with parabolae. (In the Euclidean case it's contact with circles and lines.) *Be bold*: you have no idea where your ideas will lead you. – Fly by Night Dec 25 '12 at 20:56
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    Basically I am already amazed to where I got. This question is connected to http://math.stackexchange.com/questions/254620/pi-in-arbitrary-metric-spaces which is something that was created in a boring reading and is slowly taking shape. Thanks to you, I finally was able to find a quasi-norm for which I can calculate $\pi$ with mathematica, and therefore I could find a norm with $\pi\approx 42$ :D – CBenni Dec 25 '12 at 21:11
  • @CBenni Be very careful if you use the "area norm". This is only valid for curves without Euclidean inflections. In other words, we need $\dot{\gamma}$ and $\ddot{\gamma}$ to be linearly independent along the curve. (Where $\dot{\gamma}$ is differentiation of $\gamma$ with respect to a parameter *of your choice*, $t$.) If they are dependent, then we need to delete the bad points and consider the arcs. – Fly by Night Dec 25 '12 at 21:16
  • I will have to look into that a bit more closely, its too late in the evening to do that now ^.^ – CBenni Dec 25 '12 at 21:18
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    @CBenni No worries. PM me and we can continue the discussion. Merry Christmas! – Fly by Night Dec 25 '12 at 21:20