High-level convexity for products of squared Euclidean distance functions

We study smooth functions on Euclidean space whose Hessian is positive definite outside a bounded set, with emphasis on products of squared distance functions. More precisely, we first prove a simple convexity principle: if the superlevel region $f^{-1}([c,\mathrm{\infty }))$ is contained in the Hessian-positive region of $f$, then the sublevel set $\{f\le c\}$ is convex. We apply this to finite products $F_P(x)={\prod }_{p\in P}\mathrm{\parallel }x-p{\mathrm{\parallel }}^2$, proving that their Hessian-positive complements are bounded. For the two-centre product $F_{p,q}(x)=\mathrm{\parallel }x-p{\mathrm{\parallel }}^2\mathrm{\parallel }x-q{\mathrm{\parallel }}^2$ in dimension $n\ge 2$, we compute the Hessian-positive region and the exact value $$h_{max}(F_{p,q})=\frac{\mathrm{\parallel }p-q{\mathrm{\parallel }}^4}{4}\mathrm{.}$$ This value is sharp for convexity of sublevel sets in the following sense: we prove convexity above it and nonconvexity below it. This also gives the exact convexity and quasiconvexity truncation levels for the two-centre model.