A laplace duality for integration

We consider the integral v(y) = Ky f (x)dx on a domain Ky = {x $\in$ R d\,: g(x) $\le$ y}, where g is nonnegative and Ky is compact for all y $\in$ [0, +$\infty$). Under some assumptions, we show that for every y $\in$ (0, $\infty$) there exists a distinguished scalar $\lambda$y $\in$ (0, +$\infty$) such that which is the counterpart analogue for integration of Lagrangian duality for optimization. A crucial ingredient is the Laplace transform, the analogue for integration of Legendre-Fenchel transform in optimization. In particular, if both f and g are positively homogeneous then $\lambda$y is a simple explicitly rational function of y. In addition if g is quadratic form then computing v(y) reduces to computing the integral of f with respect to a specific Gaussian measure for which exact and approximate numerical methods (e.g. cubatures) are available.