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Anton Petrunin
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Let $K=K_1\cup\dots\cup K_n$ and $K_i$ are convex.

Consider the nerve $N$ of your covering $K_i$. Note that $N$ is homotopically equivalent to $K$.

Calculating homology groups of $N$ and you may get a "no" answer if you are (un)lucky.

Anton Petrunin
  • 45k
  • 14
  • 135
  • 299