  We consider several local versions of the doubling condition and Poincar\'e
inequalities on metric spaces. Our first result is that in proper connected
spaces, the weakest local assumptions self-improve to semilocal ones, i.e.
holding within every ball.
  We then study various geometrical and analytical consequences of such local
assumptions, such as local quasiconvexity, self-improvement of Poincar\'e
inequalities, existence of Lebesgue points, density of Lipschitz functions and
quasicontinuity of Sobolev functions. It turns out that local versions of these
properties hold under local assumptions, even though they are not always
straightforward.
  We also conclude that many qualitative, as well as quantitative, properties
of p-harmonic functions on metric spaces can be proved in various forms under
such local assumptions, with the main exception being the Liouville theorem,
which fails without global assumptions.
