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Problems in Euclidean Space: Application of Convexity

1 rating: 3.0
A book by H. G. Eggleston

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 … see full wiki

Author: H. G. Eggleston
Publisher: Dover Publications
1 review about Problems in Euclidean Space: Application...

A set of complex problems in real analysis are solved in detail

  • Jun 23, 2010
Rating:
+3
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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