If a non-constant convex function has a global minimum, is it necessarily unique?

Question

Let $f : \mathbb{R}^n \to \mathbb{R}$ be a non-constant convex function. However, I do NOT assume strict convexity.

Still, if $f$ has a global minimum, is it necessarily unique?

For strictly convex cases, the result is well-known. But I wonder what would happen if the assumption "strict" is dropped (still being non-constant though..)

Answer 1

What about $f(x)=\sum\limits_{i=1}^k x_i^2$ for $1\le k<n$?

Answer 2

Here is a simple example in one dimension: $f(x) = \max\{0,x\}$.