The shapes of indifference curves are smooth with unique tangent at any bundle we have well define marginal rate of substitution (MRS). MRS is undefined if m (slope) = kinked.
1. Strictly convex: Using the example of any 2 bundles on (A & B) the indifference curve and construct a line segment between them. With the exception of A&B every bundle on the line segment lies above the IC (therefore, more is preferred). See graph below (of diminishing MRS).
2. Non Convex: 1 bundle on the line segment that is less preferred than any bundle on the IC. The result is an increase in MRS for at least 1 portion of IC. See Graph 2.
3. Weakly Convex: Part of the indifference curve is a line segment. Thus it fails to satisfy criterion for either 1 or 2. See Graph, non-increasing MRS