Strange phenomena in convex and discrete geometry. Ed. by James J. Dudziak
The field of Convex and Discrete Geometry is rich in intuitive problems which are easy to state but hard to solve, and sometimes the answers to plausible, but longstanding conjectures have been unexpected and surprising. In this exciting book, the author describes in seven chapters such ``strange phenomena'', where the solutions have been counterintuitive, or spectacular in some other way. In general, complete proofs are provided, without presupposing special knowledge, except that occasionally older results from number theory, analysis, or the theory of convex bodies have to be applied without proof. The first chapter is devoted to the famous Borsuk problem. It asks whether every bounded set in Euclidean space \(R^n\) is a union of \(n+1\) sets of smaller diameter (this question can be considered as one in convex geometry). The surprising answer (after 60 years) was that there exist finite sets in \(R^n\) for which any decomposition into sets of smaller diameter requires more than \(1.07^{\sqrt n}\) such sets. Chapter 2 treats finite packing problems. L. Fejes Tóth's sausage conjecture says that the volume of the convex hull of \(m\) nonoverlapping unit balls in \(R^n\), \(n\geq 5\), is minimal only when the centres of the balls are collinear. The chapter reproduces the proof (after almost 20 years) of the sausage conjecture in high dimensions. In contrast to this (but not so surprising, perhaps), it is shown that a convex body of minimal surface area into which \(m\) unit balls can be packed is roughly spherical when \(m\) is large. In Chapter 3, it is first proved that every convex body that tiles \(R^n\) by translations also admits a lattice tiling of \(R^n\). Then a star body is constructed that tiles \(R^{10}\), but not as a lattice. The first topic of Chapter 4, on local packing phenomena, is the existence of a convex body in \(R^n\), for large \(n\), which has smaller kissing number than a second one, but larger blocking number. Then two packing lattices for a tetrahedron are found such that the first has maximum packing density and kissing number 14, while the second has smaller density, but kissing number 18. Chapter 5 is devoted to curvature properties of convex bodies and, in particular, to the bizarre curvature behaviour of typical convex bodies, in the Baire category sense. Chapter 6 treats first the Busemann intersection inequality and then the Busemann-Petty problem: If a centered convex body in \(R^n\) has central hyperplane sections of larger \((n-1)\)-volume than another one, must it also have larger volume? A negative answer for \(n\geq 8\) is presented; the complete solution of the problem (after 38 years) is only mentioned in the notes. Also the analogous question for projections is considered. Chapter 7 gives a proof of Dvoretzky's theorem on the existence of nearly spherical central \(k\)-sections of highdimensional centered convex bodies. Each chapter puts the considered problems in their proper context and contains valuable historical remarks, in particular giving all the names of contributors to the solutions. The book should be equally fascinating for experts as for newcomers to the field.
scientific article; zbMATH DE number 917410