  This work puts forward a form finding problem of designing a least-volume
vault that is a surface structure spanning over a plane region, which via pure
compression transfers a vertically tracking load to the supporting boundary.
Through a duality scheme, developed recently for the design of pre-stressed
membranes, the optimal vault problem is reduced to a pair of mutually dual
convex problems $(\mathcal{P})$, $(\mathcal{P}^*)$ formulated on the 2D
reference region. The vault constructed upon solutions of those problems is
proved to be both of minimum volume and minimum compliance; analytical examples
of optimal vaults are given. Through a measure-theoretic approach, thus found
optimal vaults are proved to solve the Prager problem of designing a 3D
structure that by compression carries a transmissible load. The ground
structure method applied to the convex problems furnishes a pair of discrete,
conic quadratic programs $(\mathcal{P}_X)$, $(\mathcal{P}_X^*)$ leading to
optimal design of grid-shells. By adopting the member-adding adaptive technique
this pair is efficiently tackled numerically, which is demonstrated on a number
of examples where highly precise grid-shell approximations of optimal vaults
are found.
