  We develop a new approach for the estimation of a multivariate function based
on the economic axioms of quasiconvexity (and monotonicity). On the
computational side, we prove the existence of the quasiconvex constrained least
squares estimator (LSE) and provide a characterization of the function space to
compute the LSE via a mixed integer quadratic programme. On the theoretical
side, we provide finite sample risk bounds for the LSE via a sharp oracle
inequality. Our results allow for errors to depend on the covariates and to
have only two finite moments. We illustrate the superior performance of the LSE
against some competing estimators via simulation. Finally, we use the LSE to
estimate the production function for the Japanese plywood industry and the cost
function for hospitals across the US.
