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Is it possible to have preferences that is strictly increasing but not convex?

Will perfect substitutes indifference curves show strictly increasing but not convex preferences? I am confused, as won't perfect substitutes be considered as convex?

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Take $u(x,y)=x^2+y^2$ on $\mathbb R_+^2$. The function is strictly increasing in both $x$ and $y$, but the indifference curves are concave to the origin. Hence the preference it represents cannot be convex. For example, $u(1,0)=u(0,1)=1$, but the average of the two bundles is less preferred: \begin{equation} u\left(\frac12,\frac12\right)=\frac12<1=\frac12u(1,0)+\frac12u(0,1). \end{equation}

The indifference curves for perfect substitutes are straight lines. They still represent a convex preference, although not a strictly convex one.

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