Assume that $\Sigma^2$ is a closed surface in $\mathbb{R}^3$ defined by the equation $\rho(x)=1$, where $\rho$ is some smooth function so that $\nabla \rho\neq 0$. Let $A=H(\rho)$ be the Hessian matrix of $\rho$. My question arises, whether $\left<Ax,x\right>>0$ for $x\neq 0$ implies that the domain $\Omega$ bounded by $\Sigma^2$ is convex?



Hessian matrix and its positiveness
Yes. your condition implies that the surface is locally convex, and the fact that locally convex implies globally convex is [a theorem of Tietze.](https://mathoverflow.net/questions/22062/to-what-extent-is-convexity-a-local-property/22091#22091)


