Nearconvexity

Nearconvexity refers to a property of functions or sets in mathematical analysis and optimization that approximates or closely resembles convexity, yet may not strictly satisfy all conditions of convexity. The concept is primarily used in contexts where true convexity is too restrictive, and a softer, approximate version provides sufficient structure for analysis and algorithm development.

In the context of functions, a nearconvex function generally exhibits properties where the deviation from convexity is controlled or bounded. For example, a function might be considered nearconvex if the difference between its value at a convex combination of points and the convex combination of its values at those points is small or limited within a certain margin. Such functions retain many desirable characteristics of convex functions, such as approximate local optimality and certain stability properties, making them useful in non-convex optimization problems where convexity assumptions are too restrictive.

For sets, nearconvexity describes sets that are close to convex sets in a geometric sense. A nearconvex set can often be approximated or sandwiched between convex sets with a bounded "distance" or difference, allowing techniques from convex analysis to be extended or adapted.

The concept of nearconvexity has applications in economic theory, machine learning, and optimization, where it facilitates the development of algorithms that can efficiently handle problems that are "nearly convex" but exhibit non-convexities in certain regions. Its study aids in understanding the structure of complex problems and in designing approximation algorithms with provable guarantees.

Overall, nearconvexity serves as a bridge between convex and non-convex analysis, providing a framework for analyzing and solving broader classes of problems where strict convexity does not hold.