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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$\begingroup$ 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? $\endgroup$– András BátkaiCommented May 11, 2013 at 21:23
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$\begingroup$ 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 $\endgroup$– TanjaCommented May 11, 2013 at 22:34
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$\begingroup$ 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. $\endgroup$– Gerald EdgarCommented May 12, 2013 at 12:37
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$\begingroup$ 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 $R^n$). $\endgroup$– MishaCommented May 12, 2013 at 22:51
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It depends on the metric you use on $O(X)$. I guess the Hausdorff one?
answered May 11, 2013 at 21:33
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$\begingroup$ 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 $\endgroup$– TanjaCommented May 11, 2013 at 22:35
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1$\begingroup$ 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 $n$. $\endgroup$ Commented May 11, 2013 at 22:54
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$\begingroup$ No, but that has the same Lipschitz structure (and thus the same continuous functions) as the Hausdorff metric. $\endgroup$– user5810Commented May 11, 2013 at 23:27
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$\begingroup$ 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". $\endgroup$ Commented May 12, 2013 at 0:12
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$\begingroup$ 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 $\endgroup$– TanjaCommented May 12, 2013 at 0:15