## Convex Sets

I am trying to confirm these following statements:

(1) If $\succeq$ is convex, then if $y \succeq x$ and $z \succeq x$, then $ty+ (1-t)z \succeq x$.

(2) Suppose $x \sim y$. If $\succeq$ is convex, then for any $z = ty + (1-t)x$, $z \succeq x$. Basically this is saying if $x$ is indifferent to $y$, then one would prefer $z$ over $x$. I am not sure why this is the case.

A preference relation $\succeq$ is monotone if $x \succ y$ for any $x$ and $y$ such that $x_l > y_l$ for $l = 1, \ldots, L$. It is strongly monotone if $x_l \geq y_l$ for all $l = 1, \ldots, L$ and $x_j > y_j$ for some $j \in \{1, \ldots, L \}$ implies that $x \succ y$

Source: www.ksg.harvard.edu/nhm/notes2006/notes3.pdf