The main results of this paper offer sufficient conditions in order that an approximate lower Hermite–Hadamard type inequality implies an approximate Jensen convexity property. The key for the proof of the main result is a Korovkin type theorem.

A general method is given for constructing sets of sufficient linear conditions that ensure convexity of a polynomial in Bernstein–Bézier form on a triangle. Using the linear conditions, computational methods based on macro-element spline spaces are developed to construct convexity preserving splines over triangulations that interpolate or approximate given scattered data.

A real-valued set function is (additively) approximately submodular if it satisfies the submodularity conditions with an additive error. Approximate arises in many settings, especially machine learning, where evaluation might not be exact. In this paper we study how close such functions are to truly functions. We show that defined on a ground of n elements O(n2) pointwise-close function. This r...

This paper describes a general approach to compute a family of convexity measures. Inspired by the use of geometric primitives (such as circles) which are often fitted to shapes to approximate them, we use convex polygons for this task. Convex polygons can be generated in many ways, and several are demonstrated here. These polygons are scaled and translated to ensure that they fit the input sha...

In this paper, we investigate minimal (weak) approximate Hessians, and completely answer the open questions raised by V. Jeyakumar and X. Q. Yang. As applications, we first give a generalised Taylor’s expansion in terms of a minimal weak approximate Hessian. Then we characterise the convexity of a continuously Gâteaux differentiable function. Finally some necessary and sufficient optimality con...