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Non-math version of the problem: I am running with a GPS device, recording my path. I know the curve the GPS has recorded. However, the GPS device actually has an accuracy, which I can assume to be fixed (for starters).

Assuming that the curve the GPS device records is continuous and that the actual curve is also continuous (as I do not possess the power of teleportation :) how can I calculate the expected length of my run?

Now, I can translate the problem in math terms: I am looking for the average length of a function $f:[0,T]\mapsto\mathbf{R}^2$, such that $f \in C^1[0,T]$ and $\|f-g\|_{\infty}<c$, where $g \in C^1[0,T]$ is a fixed function (the recorded data), $c$ is a constant (the accuracy of the GPS) and $T$ is the time it took me to run the distance.

However, when it comes to the solution, I have absolutely no idea. I have not studied much functional analysis, so if this is a trivial exercise, sorry. Any pointers would be appreciated.

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Good question! The main issue as it stands is that the average length of such a function isn't really defined as it stands. There are (admittedly silly) paths of arbitrarily large length which satisfy your $\lVert f - g\rVert$ condition, and there's no particular reason why you shouldn't have followed any of them. You'd need some plausability reason to discard these, based on deciding how 'likely' a particular path is to be correct. – Sharkos May 6 at 0:38
@Sharkos Thank you for your comment, I had not realized this. Could you read my comment below Norbert's answer, so that I don't copy it? – K.Steff May 6 at 9:15
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That definitely fixes the existence of arbitrarily long paths since length is now bounded by $VT$, though I hope this is respected by the data given. The issue I raised still stands, however. You can only have an average of something if you have some sense how likely each particular path is. – Sharkos May 6 at 9:46

1 Answer

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If you were drunk your path may look like $f(t)=g(t)+c\sin(\omega t)$ for sufficiently large $\omega $. In this case your path $f$ will not significantly differ from $g$ in $\Vert\cdot\Vert_\infty$ metric, but the length of your path could be arrbitrary large. Hence you need to reformulate your problem in terms of $\Vert\cdot\Vert_1$ metric or put some restrictions on possible trajectories.

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Thank you for your answer. I had not thought of that, not to mention in the current form, there are even non-rectifiable curves allowed. I believe that requiring a piecewise-continuous first derivative, that is bounded by a constant $V$, is sufficient to exclude pathological cases. Is this correct? – K.Steff May 6 at 9:10
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Even in this case we have only bounds for possible length's of $f$. If you really need expected value of the length, you need to put some probability measure on the space of all possible trajectories. Since this space is the unit ball (with center $f$ and radii $c$) in the space of piecewise-continuous functions you will have problem. Becuase there is no translation invariant $\sigma$-additive measure on the unit ball of infinitedimensional space. – Norbert May 6 at 10:55
At last if you really want to get expected value I suggest you finite amount of parameters. Use them to parametrize curves $f$ (you will not cover all possible, but I think you could describe sufficiently many). Put some probability measure on the space of your parameters and compute expected value. – Norbert May 6 at 10:57
"there is no translation invariant σ-additive measure on the unit ball of infinitedimensional space" This seems quite obvious now that it is pointed out, but could you provide a reference? Also, if I say that $\|f-g\|_1$ has a normal distribution, would that be enough for a well-defined problem? – K.Steff May 7 at 19:38
1. See Halmosh's Hilbert Space Problem Book, problem 12. 2. Since you have no well deined probability measure here (unexistence of "good measure" is essential property of infinite dimensions not a particular norm) – Norbert May 7 at 22:27

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