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Let O(X) be the metric space of all compact subsets of a compact set X in Rn and let L be an element of O(X). Let vol(L) be the volume of L. How do we prove that vol(L) is a continuous function on O(X)?

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closed as off topic by Bill Johnson, Goldstern, Andreas Blass, Chris Godsil, Misha May 12 '13 at 22:51

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Though I believe this question is better suited at math.stackexchange.com , let me give you a hint. How do you define your metrics on compact sets? –  András Bátkai May 11 '13 at 21:23
    
the metric is defined as d(L1,L2)=sup{d(x1,L2):x1 in L1}+sup{d(x2,L1):x2 in L2}. Here L1 and L2 are nonempty compact subsets of Rn and by distance from a vector x to the set L we mean d(x,L)=inf{d(x,y):y in L}. With this metric is not difficult to show that O(X) is a metric space but I don't know how to prove continuity of vol(L) on O(X)! Do I use the open set approach or delta-sigma def? Help! Thank you –  Tanja May 11 '13 at 22:34
    
If you mean for $L$ to be convex, add that to the text of the question. Then we can use the fact that the boundary of a convex set has measure zero. –  Gerald Edgar May 12 '13 at 12:37
    
Tanja: With the additional assumption that compact sets in question are convex, the problem has positive and elementary geometric solution (no need to use convergence of characteristic functions, all you need is volume monotonicity under inclusion applied to slightly rescaled versions of convex sets that you have) and does not belong to MO. Therefore, I am casting the last "close" vote. However, it is a good question to ask at math.stackexchange. Do not forget to add the convexity assumption: Compact sets need not be convex (think of a 2-point set in $R^n$). –  Misha May 12 '13 at 22:51

1 Answer 1

It depends on the metric you use on $O(X)$. I guess the Hausdorff one?

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the metric is defined as d(L1,L2)=sup{d(x1,L2):x1 in L1}+sup{d(x2,L1):x2 in L2}. Here L1 and L2 are nonempty compact subsets of Rn and by distance from a vector x to the set L we mean d(x,L)=inf{d(x,y):y in L}. With this metric is not difficult to show that O(X) is a metric space but I don't know how to prove continuity of vol(L) on O(X)! Do I use the open set approach or delta-sigma def? Help! Thank you – Tanja 0 secs ago –  Tanja May 11 '13 at 22:35
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This is indeed the Hausdorff distance. I'm no expert, but I think the volume is not continuous with respect to this topology. Here is my counter-example: in $\mathbb{R}^2$ consider the square $C=[0,1]\times[0,1]$, and for $n\in\mathbb{N}$ let $\Gamma_n\subset\mathbb{R}^2$ be the graph of the $n-th$ curve $\gamma_n:t\in[0,1]\to C$ in the construction of the Peano curve. It should not be to hard to show that $d(C,\Gamma_n)\to 0$, while $vol(C)=1$ and $vol(\Gamma_n)=0$ for all $n$. –  leo monsaingeon May 11 '13 at 22:54
    
No, but that has the same Lipschitz structure (and thus the same continuous functions) as the Hausdorff metric. –  Ricky Demer May 11 '13 at 23:27
    
The title of the question contains "convex", which would block leo's counterexample, but the actual text of the question doesn't. Maybe "convex" was intended to be in the question, or maybe it's just a typo and the title was intended to say "compact". –  Andreas Blass May 12 '13 at 0:12
    
In what I am reading the metric topology of O(X) is used to construct some continuous functions(which are used in the proof of Minkowski's Second theorem). And seems like the Vol(L) should be a continuous function. Isn't Vol(L) the Lebesgue Measure of L? Since L is after all a compact set in Rn i think its volume is the Lebesgue Measure of L....and that is a number! So the volume must be a function V:O(X)->R –  Tanja May 12 '13 at 0:15

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