Convex Geometry

Abstract

Convexity has an immensely rich structure and numerous applications. On the other hand, almost every "convex" idea can be explained by a two-dimensional picture. −Alexander Barvinok [28, p.vii] We study convex geometry because it is the easiest of geometries. For that reason, much of a practitioner's energy is expended seeking invertible transformation of problematic sets to convex ones. As convex geometry and linear algebra are inextricably bonded by linear inequality (asymmetry), we provide much background material on linear algebra (especially in the appendices) although a reader is assumed comfortable with [374] [376] [233] or any other intermediate-level text. The essential references to convex analysis are [230] [349]. The reader is referred to [372] [28] [442] [44] [66] [346] [405] for a comprehensive treatment of convexity. There is relatively less published pertaining to convex matrix-valued functions. [247] [234, 6.6] [335] 2.1 Convex set A set C is convex iff for all Y , Z ∈ C and 0 ≤ µ ≤ 1 µ Y + (1 − µ)Z ∈ C (1) Under that defining condition on µ , the linear sum in (1) is called a convex combination of Y and Z. If Y and Z are points in real finite-dimensional Euclidean vector space [259] [450] R n or R m×n (matrices), then (1) represents the closed line segment joining them. Line segments are thereby convex sets; C is convex iff the line segment connecting any two points in C is itself in C. Apparent from this definition: a convex set is a connected set. [294, 3.4, 3.5] [44, p.2] A convex set can, but does not necessarily, contain the origin 0. An ellipsoid centered at x = a (Figure 15 p.36), given matrix C ∈ R m×n and scalar γ B E = {x ∈ R n | |C(x − a) 2 = (x − a) T C T C(x − a) ≤ γ 2 } (2) (an ellipsoidal ball) is a good icon for a convex set. 2.1 2.1 Ellipsoid semiaxes are eigenvectors of C T C whose lengths are reciprocal square root eigenvalues. This particular definition is slablike (Figure 13) in R n when C has nontrivial nullspace.