Strictly increasing but not convex preferences

Question

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?

Answer 1

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.