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