An elementary introduction to modern convex geometry. (English) Zbl 0901.52002

Ball, Keith

Levy, Silvio (ed.), Flavors of geometry. Cambridge: Cambridge University Press. Math. Sci. Res. Inst. Publ. 31, 1-58 (1997).

Based on series of lectures given at the Introductory Workshop in Convex Geometry (Mathematical Sciences Research Institute, Berkeley 1996), the author wrote a nice article presenting a collection of central topics from convex geometry, and instead of giving a comprehensive description of the state of the art in each case, he describes the main ideas necessary for a modern view of convex geometry. The style of his writing makes these ideas accessible to a broad audience. The topics discussed are: spherical sections of the cube (approximation of the Euclidean ball by centrally symmetric polytopes), John’s ellipsoid, volume ratios and spherical sections of the \(n\)-dimensional cross polytope \(B^n_1\) (e.g., approximating the ball by \(B^n_1 \cap UB^n_1\), where \(U\) is an orthogonal transformation), the Brunn-Minkowski inequality and its extensions, the reverse isoperimetric problem (successfully investigated by the author himself), the central limit theorem and large deviation inequalities (including Bernstein’s inequality, also called Hoeffding’s inequality by probabilists), concentration of measure in geometry (geometric analogues of Bernstein’s deviation inequality which are closely related to approximate isoperimetric inequalities), and Dvoretzky’s theorem (understood as a generic term for statements related to the effect that high-dimensional bodies have almost ellipsoidal slices).
In a convenient and compressed manner the reader is led to understand the main developments and central results close to the topics described above. By a lot of nice figures the author shows the geometric nature of many discussed problems, strongly facilitating the reader’s approach.
For the entire collection see [Zbl 0882.00019].

Reviewer: H.Martini (Chemnitz)

MSC:

52-02

Research exposition (monographs, survey articles) pertaining to convex and discrete geometry

Keywords:

Dvoretzky’s theorem; Brunn-Minkowski inequality; spherical sections; John’s ellipsoid; reverse isoperimetric inequality; central limit theorem; normed spaces; convex geometry; cube; simplex; cross polytope; isoperimetric problem; Prékopa-Leindler inequality; Busemann’s theorem