zbMATH — the first resource for mathematics

Geometric measure theory. A beginner’s guide. 3rd ed. (English) Zbl 0974.49025
San Diego, CA: Academic Press. x, 226 p. (2000).
The first (1988) and the second (1995) editions of this book have been reviewed in Zbl 0819.49024 and Zbl 0671.49043, respectively. The third edition includes four new chapters beyond those in the second edition. These new chapters are as follows: Chapter 14. Proof of the Double Bubble Conjecture. This chapter outlines the latest proofs that the standard double bubble is the least-perimeter way to enclose and separate two prescribed volumes. Chapter 15. The Hexagonal Honeycomb and Kelvin Conjectures. The hexagonal honeycomb conjecture is that the tiling by regular hexagons is the least perimeter way to partition the plane into equal areas. Though known since antiquity, this conjecture was not fully proved until 1999. A related question of how to ideally partition three dimensional space led Lord Kelvin to make an elegant conjecture as to the least area partitioning. Lord Kelvin’s conjecture was disproved by D. Weaire and R. Phelan 100 years later in 1994 [Phil. Mag. Lett. 69, No. 2, 107-110 (1994; Zbl 0900.52003)]. This chapter includes a proof of the hexagonal honeycomb conjecture and describes Lord Kelvin’s conjecture and the more efficient partitioning of \({\mathbb R}^3\) discovered by Weaire and Phelan. Chapter 16. Immiscible Fluids and Crystals. This mainly descriptive chapter discusses models of systems of immiscible fluids and of crystals. Chapter 17. Isoperimetric Theorems in General Codimensions. This chapter provides a brief sketch of Almgren’s proof of the general isoperimetric inequality in Euclidean spaces [F. J. Almgren, Indiana Univ. Math. J. 35, 451-547 (1986; Zbl 0585.49030)].

49Q15 Geometric measure and integration theory, integral and normal currents in optimization
49-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to calculus of variations and optimal control
49Q05 Minimal surfaces and optimization
49Q20 Variational problems in a geometric measure-theoretic setting
53-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to differential geometry
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)