Minimum Phase Means Fastest Decay
The previous example is an instance of the following general result:
That is, the signal energy in the first samples of the minimum-phase case is at least as large as any other causal signal having the same magnitude spectrum. (See  for a proof outline.) Thus, minimum-phase signals are maximally concentrated toward time 0 when compared against all causal signals having the same magnitude spectrum. As a result of this property, minimum-phase signals are sometimes called minimum-delay signals.
Maximum Phase Filters