As a practical application of this, consider 1-d optimization using some kind of stochastic, "jumping" process. So long as the global minimum is in a kind of l

As a practical application of this, consider 1-d optimization using some kind of stochastic, "jumping" process. So long as the global minimum is in a kind of larger well then the information concerning it's location is "global" enough for the jumping process to work its way down there. If the well the global minimum lives in is sufficiently small—if it is a sufficiently local event—then your stochastic search is bound to fail. At the limit, you have a discontinuity, a global minimum which literally has no global presence and is essentially impossible to find (you can show expected convergence times going to infinity as the minimum looks more and more discontinuous). Fortunately, few processes are genuinely discontinuous and they usually have some kind of non-local "well of attraction". But in higher and higher dimensions, these wells can be made to appear nearly infinitely narrow. All together this enforces the principle of the curse of dimensionality. High dimensionality hurts you in statistics and search unless every dimension is just chock full of signal.