Determining if a function of two variables is convex

Question

We are told that a function is convex only if the following inequality holds:

f(tx + (1 - t) x' , ty + (1 - t) y') ≤ tf(x, y) + (1-t) f(x', y')

for 0 ≤ t ≤ 1 and all pairs of points (x, y) and (x', y').

What does this mean geometrically?

Answer 1

Draw the graph of a convex function and pick two points on the graph. The line interpolating these two points lies above the graph.