Why is sum a succinct constraint?
We can show that the "sum above threshold" is not succinct by providing a counter example.

As you wrote the definition is


  Given A1, the set of items satisfying a succinctness constraint C, then any set S satisfying C is based on A1 , i.e., S contains a subset belonging to A1


Hence, we can provide a counter example by providing a set A1 satisfying the constraint while none of its subsets satisfying it.

Consider three items, a,b and c such that each one on them costs 1.
Let the constraint C be sum(S.Price) &gt;= 3

For the set {a, b, c} the sum of prices is 3 and therefore the constraint C is satisfied.
For each of the subsets of {a, b, c} the sum of prices is lower than 3 and therefore C is not satisfied.
We found a counter example in which a set satisfies a "sum above threshold" while none of its subsets satisfies it. Hence, "sum above threshold" is not succinct.