Name:
Siwei Shen
Nationality:
Chinese
College:
Science College
Degree:
Bachelor of Science
Major(s):
Computer Science
Mathematics
Email:
Dase35@aol.com
Tel. No.
622,4523
Final Report
Proposal for the Project: Convexity at a Point
The main topic of the project I am going to pursue next semester is convex functions.
The study of convex functions begins in the context of real-valued functions of a real variable. A rich diversity of pertinent theorems can be found having an elegance that is rooted in the very simplicity of their proofs. The results, however, are not trivial but have a wide range of applications in mathematics, optimization in particular. One example of this would be the study of maxima and minima. As is known, such study is characterized by certain complexity for general functions. Convex functions, however, exhibit a particularly simply extremal structure, which makes it more possible to derive some solutions with the help of convexity theories.
Likewise, the study of convex functions plays an important role in a number of newborn areas of mathematics, areas that appear in response to the tremendous impetus received from outside mathematics, such as economic, flows in network, and so on. Under this category are game theory, linear and nonlinear programming, control theory, and many others that accompany the invention of high-speed computers, aiming at solving problems in areas mentioned above. In all these studies, convexity theory turned out to be at the core.
Before moving on to the specific problems I am to tackle, some basic terms and concepts of convexity have to be understood:
Convex function: Define a function F mapping I into R, where I is a subset of R. F is called convex if and only if,
For all x, y in I, F[cx + (1 - c)y] <= cF(x) + (1 - c)F(y), where c lies in [0, 1].
Geometrically, it means that if P, Q, and R are any three points on the graph of F with Q between P and R, then Q is on or below chord PR.
Convex set: Let U be a subset of a linear space L. U is said to be convex if and only if it contains the line segment connecting any two of its points. Mathematically, a subset U of a linear space is convex if and only if,
For all x, y in U, z = [cx + (1 - c)y] is also in U, for any c in [0, 1].
The actual project I am going to investigate will be "Convexity at a Point". To describe, let U be a fixed, not necessarily convex set in a linear space L. Call F from U to R convex at a point x in U if for each representation x = cy + (1 - c)z with y, z in U and c in [0, 1], we have F(x) <= cF(y) + (1 - c)F(z). In turn, say that F is convex on a subset V of U if it is convex at each point of V (Refer to Convex Functions, pp268).
The goal of working on this new notion of a convex function is to, if possible, determine the following items or some of them:
1. Relation between "Being convex at a point" and "Having support at a point";
2. Relation between Convexity and the validity of Jensen's Inequality at a point;
3. Conditions of subsets V of U to assure the existence of F from R to R being convex on V;
4. Conditions of subsets V of U to assure the convexity on U provided the convexity on V;
5. Development of differentiation theory characterizing this convexity theory.
Roberts, A. Wayne & Dale E. Varberg. Convex Functions. New York & London: Aademic Press, 1973.
Final Report
