Why politicians talk in code
The median voter theorem is an interesting result in the study of elections. Suppose you have some political spectrum, and two candidates. Doesn't matter how many dimensions there are to that spectrum, even, with two candidates; the politician who is closer to the center (median) of voter positions will be closer to more voters. Voters - if acting rationally - will vote for the closest candidate to them.
Voila, so suddenly we have centrist candidates - or do we? There's another problem: Turnout. The further away a voter is from a candidate, the less likely a voter is to turn out. In fact, if they feel far away on the fringe, they may feel the difference between the two centrist candidates is negligible. Perhaps a third party candidate will look attractive - it's time to make a statement!
But what if you can occupy more than one position at once? Suddenly, you can secure much more of the political spectrum. This is why politicians talk in code, and try to position their opponents as extremists; why we see different messages sent through different media. Coded language is understood differently by different segments of the political spectrum, and by segmenting your audience into different groups, it allows you to try to position yourself near the median of each group rather than the whole population.
And this is how a politician masters the median voter theorem; not by moving carefully to the center of the population, but by seeming to be in different places when looked at from different perspectives.