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Suppose two sets $X\cap Y\neq\emptyset$. Let $d_{H}$ denote the Hausdorff distance of two sets. Suppose $d_{H}(X_{n},X)\rightarrow 0$ and $d_{H}(Y_{n},Y)\rightarrow 0$ with $X_{n}\cap Y_{n}\neq\emptyset$ for all $n$. Suppose all $X,Y,X_{n},Y_{n}$ are closed and convex.

Are there any theorems saying that $d_{H}(X_{n}\cap Y_{n},X\cap Y)\rightarrow 0$?

Or this may not be true?

Thanks very much.

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    What is the context here? Are they subsets of a Banach space? – 2017-01-25
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    Yes, it is OK to assume that they are in Banach space. Or even stronger, we can assume they are all compact. – 2017-01-25
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    I'm asking because it is not clear what "convex" means until you specify the context. Anyway, I think Robert's answer works in just about any reasonable context. – 2017-01-25

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Consider the following case in the plane.

$X_n$ is the triangle with vertices $[-1,0], [0,0], [-1/n,1]$,

$Y_n$ is the triangle with vertices $[1,0], [0,0], [1/n, 1]$.

$X$ is the triangle with vertices $[-1,0],[0,0],[0,1]$.

$Y$ is the triangle with vertices $[1,0],[0,0],[0,1]$.

This satisfies the hypotheses, but $X_n \cap Y_n$ is the single point $[0,0]$ while $X \cap Y$ is the line segment from $[0,0]$ to $[0,1]$.