Question on convexity

Question

If I have a function $f:\mathbb{R}^n\to\mathbb{R}$ that is convex in ${\bf x} = (x_1,x_2,\ldots,x_n)$ and strictly convex in one of the variables, say $x_1$, then is $f({\bf x})$ strictly convex in ${\bf x}$? If so, how would I prove this?

Answer 1

Take any convex (but not strictly convex) function $c:\mathbb{R} \to \mathbb{R}$ and any strictly convex function $s:\mathbb{R} \to \mathbb{R}$, then the function $f:\mathbb{R}^2 \to \mathbb{R}$ given by $f(x,y) = c(x)+s(y)$ satisfies the criteria in the question, but clearly $f$ is not strictly convex for any fixed $y$.

Answer 2

No, just consider $f(x) = x_1^2$. This is strictly convex in $x_1$ and weakly convex in $x$ but not strictly convex in $x$.