  This work proves rigorous results about the vanishing-mass limit of the
classical problem to find a shape with minimal elastic compliance. Contrary to
all previous results in the mathematical literature, which utilize a soft mass
constraint by introducing a Lagrange multiplier, we here consider the hard mass
constraint. Our results are the first to establish the convergence of
approximately optimal shapes of (exact) size $\varepsilon \downarrow 0$ to a
limit generalized shape represented by a (possibly diffuse) probability
measure. This limit generalized shape is a minimizer of the limit compliance,
which involves a new integrand, namely the one conjectured by Bouchitt\'e in
2001 and predicted heuristically before in works of Allaire & Kohn and Kohn &
Strang from the 1980s and 1990s. This integrand gives the energy of the limit
generalized shape understood as a fine oscillation of (optimal)
lower-dimensional structures. Its appearance is surprising since the integrand
in the original compliance is just a quadratic form and the non-convexity of
the problem is not immediately obvious. In fact, it is the interaction of the
mass constraint with the requirement of attaining the loading (in the form of a
divergence-constraint) that gives rise to this new integrand. We also present
connections to the theory of Michell trusses, first formulated in 1904, and
show how our results can be interpreted as a rigorous justification of that
theory on the level of functionals in both two and three dimensions, finally
settling this long-standing open problem. Our proofs rest on compensated
compactness arguments applied to an explicit family of (symmetric)
div-quasiconvex quadratic forms, computations involving the Hashin-Shtrikman
bounds for the Kohn-Strang integrand, and the characterization of limit
minimizers due to Bouchitt\'e & Buttazzo.
