  The operational meaning of coherence measure lies at very heart of the
coherence theory. In this paper, we provide an operational interpretation for
geometric coherence, by proving that the geometric coherence of a quantum state
is equal to the minimum error probability to discriminate a set of pure states
with von Neumann measurement. On the other hand, we also show that a task to
ambiguously discriminate a set of linearly independent pure states can be also
regards as a problem of calculating geometric coherence. That is, we reveal an
equivalence relation between ambiguous quantum state discrimination and
geometric coherence. Based on this equivalence, moreover, we improve the upper
bound for geometric coherence and give the explicit expression of geometric
coherence for a class of states. Besides, we establish a complementarity
relation of geometric coherence and path distinguishability, with which the
relationship between $l_1$-norm of coherence and geometric coherence is
obtained. Finally, with geometric coherence, we study multiple copies quantum
state discrimination and give an example to show how to discriminate two pure
states.
