A statement for convex sets

Question

The following statement is true or false?

Given a convex set $S$ then for any $y \in S$ and $\theta\in[0,1], \theta \in \mathbb R$ there exist $y_1,y_2 \in S, y_1 \ne y, y_2 \ne y$ such that $y=\theta y_1+(1-\theta) y_2.$

Answer 1

You don't even need $S$ convex: just take $y_1 = y_2 = y$.

Answer 2

The statement is clearly false: just think about a vertex of a square!

By the way, you guys got it wrong: by definition the empty set and a single point are convex set. The latter is indeed an affine set of dimension 0.

A set $S$ is convex iff

$$ \forall x,y \in S \Rightarrow \theta x + (1-\theta)y \in S, \quad \forall \theta \in [0,1]$$

an it doesn't say that the two point must be different. Otherwise a square would not be a convex set.