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Let $A \in \mathbb{R}^{n \times n}$ be a given symmetric and positive semidefinite matrix and consider the set $\Omega:=\left\{x \in \mathbb{R}^{n}: x^{\top} A x \leq 0\right\} .$ Show that the set $\Omega$ is convex.

My incomplete proof:

$\Omega:=\left\{x \in \mathbb{R}^{n}: x^{\top} A x = 0\right\} .$ Assume $y_1, \, y_2 \in \Omega$, prove that $\lambda y_1 + (1 - \lambda)y_2 \in \Omega$

Finally, I got $(\lambda y_1 + (1 - \lambda)y_2)^TA(\lambda y_1 + (1 - \lambda)y_2) = \lambda (1 - \lambda)(y_1^TAy_2 + y_2^TAy_1)$.

But I do not know what to do next. Can anybody help me?

amWhy
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Yanglin Zhang
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    Level sets of convex functions are convex. – Tobsn Jul 12 '21 at 19:41
  • this is a special case of https://math.stackexchange.com/questions/946156/proving-convexity-of-a-function-whose-hessian-is-positive-semidefinite-over-a-co – daw Jul 13 '21 at 06:22

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