Starting with Cook's pioneering work on NP-completeness in 1970, polynomial complexity theory, the study of polynomial-time com putability, has quickly emerged as the new foundation of algorithms. On the one hand, it bridges the gap between the abstract approach of recursive function theory and the concrete approach of analysis of algorithms. It extends the notions and tools of the theory of computability to provide a solid theoretical foundation for the study of computational complexity of practical problems. In addition, the theoretical studies of the notion of polynomial-time tractability some times also yield interesting new practical algorithms. A typical exam ple is the application of the ellipsoid algorithm to combinatorial op timization problems (see, for example, Lovasz [1986]). On the other hand, it has a strong influence on many different branches of mathe matics, including combinatorial optimization, graph theory, number theory and cryptography. As a consequence, many researchers have begun to re-examine various branches of classical mathematics from the complexity point of view. For a given nonconstructive existence theorem in classical mathematics, one would like to find a construc tive proof which admits a polynomial-time algorithm for the solution. One of the examples is the recent work on algorithmic theory of per mutation groups. In the area of numerical computation, there are also two tradi tionally independent approaches: recursive analysis and numerical analysis."

Complexity theory of real functions

On the Complexity of Convex Hulls of Subsets of the Two-Dimensional Plane

After some general introductory remarks on approximation in convex geometry we present in the main part of this article asymptotic results on best approximation of convex bodies as the number of vertices, resp. facets of the approximating poly topes tends to infinity. Tools are from (affine) differential geometry. Since the transparent geometric situation in the plane admits much more precise results, this case is treated separately. In higher dimensions the relations to Dirichlet- Voronoi and Delone tilings and to the ball covering problem are indicated. Then algorithmic and, in particular, step-by-step approximation results are discussed. Here tools from number theory are applied. In the material presented emphasis is on the ideas underlying the proofs. Supplementing earlier surveys, the last chapter contains a summary of recent results on approximation of convex bodies.

Approximation by Convex Polytopes

We study first and second derivatives of computable convex functions on \(\mathbb {R}^n\). The main result of the paper is an effective form of Aleksandrov’s Theorem: we show that computable randomness implies twice-differentiability of computable convex functions.

Randomness and Differentiability of Convex Functions

Publisher Summary This chapter provides a bibliography for recursive analysis and recursive topology. Recursive mathematics investigates the “constructive” nature of mathematical results when “constructive” is interpreted via recursive function theory or Turing computability, whereas the classical laws of logic are adopted intact. Recursive analysis focuses this approach of study to classical analysis. The chapter describes some of the more recent approaches to recursive analysis, including sequence approach, filter approach, and representation approach. The sequence approach to computability has been inspired by the observation that a real function (defined on the unit interval) is computable in the sense of Grzegorczyk/Lacombe if and only if it maps computable sequences to computable sequences and admits a computable modulus of continuity. In the filter approach, the objects of study are not points, they are neighborhoods. The approach explains the existing phenomena on pathology of domains of recursive real functions while presenting new results and machinery to apply to recursive analysis. In the representation approach, classical theorems can be expressed in highly effective versions. It provides a direct way to investigate complexity in analysis.

Chapter 12 A bibliography of recursive analysis and recursive topology

In this paper,we study the computational complexity of the integrals and the derivatives of
convex functions defined on the interval [0,1].

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Papers

Computational complexity of integration and differentiation of convex functions