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?
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1Convex functions have convex level sets – Yoav Kallus Feb 23 '20 at 21:28
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Yes. your condition implies that the surface is locally convex, and the fact that locally convex implies globally convex is a theorem of Tietze.
Igor Rivin
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