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I have two variables, $x$ and $y$, and a few inequalities of the form $f(x,y) \le g(x,y)$.

I want to know if the intersection of all $(x,y)$ that satisfy each inequality is convex. Is there some generic way to do it? Maybe based on second order derivative (or the Hessian in this case), similarly to the test whether a function is convex?

Finding whether one inequality defines convex set is also good, because if they all define convex sets, then their intersection must define a convex set as well.

Thanks.

  • 3
    If the Hessian of $f$ is positive definite, then the set defined by the inequality $f(x,y)\leq c$ is convex for any constant $c$.2011-05-31
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    By definition: a set $X \subset \mathbb{R}^n$ is convex if for all $x, y \in X$ all their affine combinations $\theta x + (1 - \theta) y$ (where $0 < \theta < 1$) are also in $X$. For closed $X$ it is sufficient to show that $\frac{1}{2}(x + y) \in X$.2011-06-30
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    Never forget to try out some random sampling. It can save you a lot of time and effort if you find a counter-example.2011-08-29

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