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

Consider the [nerve][1] $N$ 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.

  [1]: http://en.wikipedia.org/wiki/Nerve_of_a_covering