Let [InlineEquation not available: see fulltext.], let [InlineEquation not available: see fulltext.] stand for a, b, c, d∈ Z_{⩾ 0} such that ad- bc= 1. Define [Equation not available: see fulltext.]In other words, we consider the sum of the powers of the triangle inequality defects for the lattice parallelograms (in the first quadrant) of area one.We prove that [InlineEquation not available: see fulltext.] converges when s> 1 and diverges at s= 1 / 2. We also prove that ∑(a,b,c,d)1(a+c)2(b+d)2(a+b+c+d)2=13,and show a general method to obtain such formulae. The method comes from the consideration of the tropical analogue of the caustic curves, whose moduli give a complete set of continuous invariants on the space of convex domains.