Selected Topics in Convex Geometry [electronic resource] / by Maria Moszyńska.

I -- Metric Spaces -- Subsets of Euclidean Space -- Basic Properties of Convex Sets -- Transformations of the Space Kn of Compact Convex Sets -- Rounding Theorems -- Convex Polytopes -- Functionals on the Space Kn. The Steiner Theorem -- The Hadwiger Theorems -- Applications of the Hadwiger Theorems -- II -- Curvature and Surface Area Measures -- Sets with positive reach. Convexity ring -- Selectors for Convex Bodies -- Polarity -- III -- Star Sets. Star Bodies -- Intersection Bodies -- Selectors for Star Bodies.

The field of convex geometry has become a fertile subject of mathematical activity in the past few decades. This exposition, examining in detail those topics in convex geometry that are concerned with Euclidean space, is enriched by numerous examples, illustrations, and exercises, with a good bibliography and index. The theory of intrinsic volumes for convex bodies, along with the Hadwiger characterization theorems, whose proofs are based on beautiful geometric ideas such as the rounding theorems and the Steiner formula, are treated in Part 1. In Part 2 the reader is given a survey on curvature and surface area measures and extensions of the class of convex bodies. Part 3 is devoted to the important class of star bodies and selectors for convex and star bodies, including a presentation of two famous problems of geometric tomography: the Shephard problem and the Busemann-Petty problem. Selected Topics in Convex Geometry requires of the reader only a basic knowledge of geometry, linear algebra, analysis, topology, and measure theory. The book can be used in the classroom setting for graduates courses or seminars in convex geometry, geometric and convex combinatorics, and convex analysis and optimization. Researchers in pure and applied areas will also benefit from the book.