Is $xy\leq z$ or $xy\geq z$ quasi convex in some infinite subset with positive measure of $\mathbb R_{\geq 0}^3$?
If not is it quasi-concave in some infinite subset with positive measure of $\mathbb R_{\geq 0}^3$?
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What is your definition of "quasi convex"? – gerw Jan 22 '20 at 12:33
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@gerw The standard one in usage. – Turbo Jan 22 '20 at 14:38
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Please give the definition. The standard definition is for a function, but in my opinion "$xy \le z$ or $xy \ge z$" is not a function. – gerw Jan 22 '20 at 14:42
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Both are neither in whole of $\mathbb R^3$. – Turbo Jan 22 '20 at 16:54
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If you do not answer my request properly, I cannot help you... – gerw Jan 23 '20 at 09:15
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@gerw $x^2\leq y$ is a convex constraint on whole of $\mathbb R^2$ and thus convex on any convex subset of $\mathbb R^2$. We cannot say same about $xy\leq z$ or $xy\geq z$ as these are not convex constraints on whole of $\mathbb R^3$. By convex constraint inequality we mean the inequality defines a convex region in $\mathbb R^n$ and the answer to my problem is positive in a region of $0$ measure (low dimensional subsets for example $y=1$). – Turbo Jan 23 '20 at 09:54