Here's an example of how to show that a geometric region, in this case, a disk, is convex.
In[12]:= Resolve[
ForAll[
{x1, y1, x2, y2},
0 <= k <= 1 &&
Element[{x1, y1}, Disk[]] &&
Element[{x2, y2}, Disk[]],
Element[k {x1, y1} + (1 - k) {x2, y2}, Disk[]]
], Reals]
Out[12]= True
