The following definition of convexity is used in this book.
A subset X of R2 or R3 is convex if whenever x and y are in X the segment connecting them is also in X.
Using this definition, a series of problems are presented and solved. Organized by chapter, they are
Chapter 1) Problems in which convexity is used either by analogy or for subsidiary arguments *) The intersection of connected open sets *) Approximations to homomorphisms of R2 onto itself *) On the projection of a plane set of finite linear measure
Chapter 2) Problems which can be reduced to problems on convex sets *) Covering a three-dimensional set with sets of smaller diameter
Chapter 3) Problems on convex sets *) Approximation to plane convex sets *) Geometrical properties for which triangles are the extremal convex curves
Chapter 4) Problems concerned with the structure of subclasses of the class of convex sets *) The asymmetry of curves of constant width *) Sets of constant width contained in a set of given minimal width *) Extremal properties of triangles circumscribing plane convex sets *) On the closest packing by equilateral triangles
Each of the problems is dealt with in detail with theorems followed by proofs. A remark summarizing the problem occurs after every problem is resolved. In terms of difficulty, some real analysis background is necessary to understand the work.
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This text for advanced undergraduates and graduate students examines problems concerning convex sets in real Euclidean spaces of 2 or 3 dimensions. It illustrates the different ways in which convexity can enter into the formulation as the solution to different problems in these spaces. 1957 edition.