Point of convexity ($g(v)\ge g(v_0)+g^{\prime }(v_0)(v-v_0)$)

For a differentiable real function $g$, a point of convexity is a point $v_0$ whose tangent line lies below the entire graph. The function need not be convex at other points. The Weierstrass excess function is exactly the difference between the function and this tangent line when applied to a variational integrand as a function of slope.