Near convexity, metric convexity, and convexity
نویسنده
چکیده
It is shown that a subset of a uniformly convex normed space is nearly convex if and only if its closure is convex. Also, a normed space satisfying a mild completeness property is strictly convex if and only if every metrically convex subset is convex. 1 Classical and constructive mathematics The arguments in this paper conform to constructive mathematics in the sense of Errett Bishop. This means roughly that they do not depend on the general law of excluded middle. More precisely, the arguments take place in the context of intuitionistic logic. Arguments in the context of ordinary logic will be referred to as classical. As intuitionistic logic is a fragment of ordinary logic, our arguments should be valid from a classical point of view, although some of the maneuvers to avoid invoking the law of excluded middle may seem puzzling. I had intended to write this paper primarily to be read classically, at least in its positive aspects, allowing constructive mathematicans to see for themselves that the arguments were constructively valid. But for those classical mathematicians who want to follow some of the constructive ne points, I include here (at the suggestion of the referee) two constructive principles about real numbers that are used instead of the classical trichotomy, a < b or a = b or a > b, which is not constructively valid.
منابع مشابه
Convexity and Geodesic Metric Spaces
In this paper, we first present a preliminary study on metric segments and geodesics in metric spaces. Then we recall the concept of d-convexity of sets and functions in the sense of Menger and study some properties of d-convex sets and d-convex functions as well as extreme points and faces of d-convex sets in normed spaces. Finally we study the continuity of d-convex functions in geodesic metr...
متن کاملConvexity and Osculation in Normed Spaces
Constructive properties of uniform convexity, strict convexity, near convexity, and metric convexity in real normed linear spaces are considered. Examples show that certain classical theorems, such as the existence of points of osculation, are constructively invalid. The methods used are in accord with principles introduced by Errett Bishop.
متن کاملBall Versus Distance Convexity of Metric Spaces
We consider two different notions of convexity of metric spaces, namely (strict/uniform) ball convexity and (strict/uniform) distance convexity. Our main theorem states that (strict/uniform) distance convexity is preserved under a fairly general product construction, whereas we provide an example which shows that the same does not hold for (strict/uniform) ball convexity, not even when consider...
متن کاملCalculating Cost Efficiency with Integer Data in the Absence of Convexity
One of the new topics in DEA is the data with integer values. In DEA classic models, it is assumed that input and output variables have real values. However, in many cases, some inputs or outputs can have integer values. Measuring cost efficiency is another method to evaluate the performance and assess the capabilities of a single decision-making unit for manufacturing current products at a min...
متن کاملSOME PROPERTIES FOR FUZZY CHANCE CONSTRAINED PROGRAMMING
Convexity theory and duality theory are important issues in math- ematical programming. Within the framework of credibility theory, this paper rst introduces the concept of convex fuzzy variables and some basic criteria. Furthermore, a convexity theorem for fuzzy chance constrained programming is proved by adding some convexity conditions on the objective and constraint functions. Finally,...
متن کامل