BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY

The author writes in the preface: “Discrete Convex Analysis is aimed at establishing a novel theoretical framework for solvable discrete optimization problems by means of a combination of the ideas in continuous optimization and combinatorial optimization.” Thus the reader may conclude that the book presents a new theory (the name “discrete convex analysis” was, apparently, coined by the author). The reader may also notice the adjective “solvable” attached to “discrete optimization problems” and hence ask whether “solvable” means solvable in principle, solvable by the new theory, solvable by all other known approaches, or solvable by some of the known approaches for which the theory provides a unified framework. I think that the best fit is given by the last option. Thus, I would like to describe the general area of discrete convexity and the contribution of the book. A set A ⊂ R is called convex if, for any two points x, y ∈ A, the interval [x, y] = { αx + (1 − α)y : 0 ≤ α ≤ 1 } also lies in A. A function f : R −→ R is called convex provided that for every λ ∈ R, the set { x : f(x) ≤ λ } is convex. Equivalently, f is convex if and only if f ( αx + (1 − α)y ) ≤ αf(x) + (1 − α)f(y) for all x, y ∈ R and all 0 ≤ α ≤ 1. These remarkably simple definitions lead to a remarkably rich and useful theory with a great many applications. Here we are interested in optimization problems with convex objective functions. Convex functions and convex sets have some nice properties as far as optimization is concerned. A local minimum of a convex function f on a convex set A is necessarily a global minimum. There is also a powerful duality theory, which we sketch below, having optimization in mind. Let 〈·, ·〉 be the standard scalar product in R. For a non-empty set A ⊂ R, let A◦ = { c ∈ R : 〈c, x〉 ≤ 1 for all x ∈ A } be the polar of A. Then, A◦ is a closed convex set containing the origin, and if A is itself a closed convex set containing the origin, then (A◦) = A (the Bipolar Theorem). For a set A ⊂ R, let [A] : R −→ R be its indicator: [A](x) = 1 if x ∈ A and [A](x) = 0 if x / ∈ A. The polarity correspondence A 7−→ A◦ preserves linear dependencies among indicators of closed convex sets: if ∑m i=1 αi[Ai] = 0 for real numbers αi and non-empty closed convex sets Ai, then ∑m i=1 αi[A ◦ i ] = 0. With every set A ⊂ R, we associate two problems. First, the Membership Problem: given a point x ∈ R, decide whether x ∈ A. Second, the Optimization Problem: given a vector c ∈ R, compute the minimum (maximum) value of the linear function 〈c, x〉 for x ∈ A. The polarity correspondence A ←→ A◦ naturally gives rise to the correspondence between the Optimization Problem for A and the Membership Problem for A◦. The Bipolar Theorem implies a certain symmetry between the Membership and Optimization problems. Suppose now that a convex body (a convex compact set with a non-empty interior) A ⊂ R is defined by its Membership Oracle, that is, a black box which, given