  Methods for measuring convexity defects of compacts in R^n abound. However,
none of the those measures seems to take into account continuity. Continuity in
convexity measure is essential for optimization, stability analysis, global
optimality, convergence analysis, and accurate modelling as it ensures
robustness and facilitates the development of efficient algorithms for solving
convex optimization problems. This paper revisits the axioms underlying
convexity measures by enriching them with a continuity hypothesis in
Hausdorff's sense. Having provided the concept's theoretical grounds we state a
theorem underlining the necessity of restricting ourselves to non-point
compacts. We then construct a continuous convexity measure and compare it to
existing measures.
