Concentration of measure is a fundamental tool in high dimensional geometry and in modern probability. However, typically it fails to capture the behavior of high dimensional objects (like norms) when fine estimates are required, such as small deviation inequalities or small ball probabilities. However the convexity (or concavity) of the function can be used to get improved estimates. Applications to convex geometry will be presented. (This talk will be based on joint works with P. Pivovarov and P. Valettas)