There is an analogous notion of thin position, defined for knots, for Heegaard splittings. The complexity of a connected surface "S", "c(S)", is defined to be formula_22; the complexity of a disconnected surface is the sum of complexities of its components. The complexity of a generalized Heegaard splitting is the multi-set "{c(S_i)}", where the index runs over the Heegaard surfaces in the generalized splitting. These multi-sets can be well-ordered by lexicographical ordering (monotonically decreasing). A generalized Heegaard splitting is thin if its complexity is minimal.