Say that $X$ and $Y$ are two convex regions in the plane, and suppose that $X \subset Y$. Further suppose that $Y$ is partitioned into disjoint convex subsets $Y_1 ,\dots, Y_n$. Is there a way of using $Y_1,\dots,Y_n$ to induce a partition of $X$ that also consists of $n$ convex subsets, call it $X_1,\dots,X_n$, such that all $X_i$ are non-empty? Obviously, if we just defined $X_i = X \cap Y_i$, it could happen that some subsets would be empty.
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$\begingroup$ Does "convex partition" simply mean each part is convex? And, what do you want out of this partition? Should it be natural, or can it be artificial? For instance, I could just slice $X$ into $n$ parallel slices and call it my partition, but I suspect you mean something when you write "induce" and "using $Y_1,\ldots,Y_n$". What is it you mean? $\endgroup$– Yoav KallusMay 20, 2015 at 3:40
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$\begingroup$ Thanks, made a clarification; it should be that each of the subsets $Y_i$ is itself convex (and similarly for the $X_i$'s). Yes, you could slice $X$ into parallel slices as you describe, but I'm trying to find some "canonical" way to obtain the partition of $X$ from the partition of $Y$. I realize that this is vague as I've stated it, and the truth is that I do not have a concrete description of what kind of shape properties I want to preserve from one to the other. $\endgroup$– Tom SolbergMay 20, 2015 at 4:57
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$\begingroup$ $X$ could be empty or have $n-1$ points... You need some more conditions. $\endgroup$– DirkMay 20, 2015 at 5:01
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$\begingroup$ How could $X$ have $n-1$ points? It's a convex region in the plane. $\endgroup$– Tom SolbergMay 20, 2015 at 5:29
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$\begingroup$ Err, sorry - of course, nonempty $X$ it should be… $\endgroup$– DirkMay 20, 2015 at 20:41
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