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