## Convex Sets

I am trying to confirm these following statements:

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

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

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

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