1. Basic convexity 2. Boundary structure 3. Minkowski addition 4. Curvature measure and quermass integrals 5. Mixed volumes 6. Inequalities for mixed volumes 7. Selected applications Appendix.

/pdf/convex-bodies-the-brunn-minkowski-theory-3raynvod45.pdf

Convex bodies : the Brunn-Minkowski theory

a ) p. 131 The discussion between eqs. (5.14) and (5.15) is incorrect (dA should be made as large as possible!). b ) p. 256 In the figure, the numbers 6) and 7) occur twice. c ) p. 292 At the end of section 12.5, it should be the space of isospectral 0Hermitian matrices. d ) p. 306 A ”Tr” is missing in eq. (13.43). e ) p. 327, Eq. (14.64b) is 〈Trρ〉B = N(14N+10) (5N+1)(N+3) should be 〈Trρ〉B = 8N+7 (N+2)(N+4)

Geometry of Quantum States: Frontmatter

Morse Theory for Cell Complexes

https://chaos.if.uj.edu.pl/~karol/erratum10.pdf

Geometry of Quantum States

Geometry of quantum states: an introduction to quantum entanglement by Ingemar Bengtsson and Karol Zyczkowski

In this paper we prove that every closed polyhedral surface in Euclidean three-space can be approximated (uniformly with respect to the Hausdorff metric) by smooth surfaces of the same topological type such that not only the (Gaussian) curvature but also the absolute curvature and the absolute mean curvature converge in the measure sense. This gives a direct connection between the concepts of total absolute curvature for both smooth and polyhedral surfaces which have been worked out by several authors, particularly N. H. Kuiper and T. F. Banchoff.

Smooth approximation of polyhedral surfaces regarding curvatures

Polyhedral maps

We construct a triangulated Mobius strip with 9 vertices which is not embeddable into R3 such that all edges are straight line segments. It even cannot be immersed polyhedrally into R3. For every integer g 2 1 it is still an open problem whether there exists a triangulated closed orientable two-dimensional manifold of genus g which is not embeddable into R3 such that all triangles are planar triangles with straight edges (cf. [1, 2]). It is natural to ask the corresponding question for nonorientable two-dimensional manifolds with boundary. Theorems 1 and 2 (and the remark) answer that question completely also for immersions. THEOREM 1. There exists a triangulated Mbbius strip with 9 vertices which is not embeddable into R3 such that all edges are straight line segments. PROOF. (1) Let I and J be disjoint closed polygonal curves in R3 with three vertices. Then I lk(I, J) I 2. (3) The triangulated Mobius strip given in Figure 2a cannot be embedded into R3 with straight edges, since for the triangular curves I = 1231 and J' = 4564 we have I lk(I, J') I> 2 by (2), (choose J = 7897), which is impossible with straight edges by (1). In Figure 2b we give another representation of the same triangulation (1 2 3 is not a triangle of the triangulation). Figure 2a presents the Mobius strip as a rectangle with a pair of opposite edges identified with a twist and Figure 2b presents it as a real projective plane with a triangle removed. DEFINITION. Let us define a polyhedral immersion of a two-dimensional simplicial complex to be an immersion such that all triangles are planar and all edges are straight line segments. Received by the editors November 17, 1982 and, in revised form, May 3, 1983. 1980 Mathematics Subject Classification. Primary 57Q35; Secondary 52A25, 57M20. (D1 983 American Mathematical Society 0002-9939/83 $1.00 + $.25 per page 519 This content downloaded from 157.55.39.177 on Wed, 16 Nov 2016 04:23:00 UTC All use subject to http://about.jstor.org/terms

/pdf/a-nonpolyhedral-triangulated-mobius-strip-eq10tjea8p.pdf

A nonpolyhedral triangulated Möbius strip

We investigate congruence classes and direct congruence classes of m-tuples in the complex projective space ℂP

 n
 . For direct congruence one allows only isometries which are induced by linear (instead of semilinear) mappings. We establish a canonical bijection between the set of direct congruence classes of m-tuples of points in ℂP

 n
 and the set of equivalence classes of positive semidefinite Hermitean m×m-matrices of rank at most n+1 with 1's on the diagonal. As a corollary we get that the direct congruence class of an m-tuple is uniquely determined by the direct congruence classes of all of its triangles, provided that no pair of points of the m-tuple has distance π/2. Examples show that the situation changes drastically if one replaces direct congruence classes by congruence classes or if distances π/2 are allowed. Finally we do the same
kind of investigation also for the complex hyperbolic space ℂH

 n
 . Most of the results are completely analogous, however, there are also some interesting differences.

Congruence criteria for finite subsets of complex projective and complex hyperbolic spaces

Publisher Summary This chapter focuses on projective geometry on modular lattices. Incidence and Order are basic concepts for a foundation of modern synthetic geometry. These concepts describe the relative location or containment of geometric objects and have led to different lines of geometry, an incidence-geometric and a lattice-theoretic one. Modularity is one of the fundamental properties of classical projective geometry. It makes projections into join-preserving mappings and yields perspectivities to be (interval) isomorphisms. It is therefore natural that order-theoretic generalizations of projective geometry are based on modular lattices and even more, the theory of modular lattices may be considered as a most general concept of projective geometry. In particular, the partially ordered set of all submodules of a module forms a (complete) modular lattice; even more general, any sublattice of the lattice of all normal subgroups of a group is a modular lattice. It considers that lattice-geometric approaches are complete geometrical structures whose geometrical objects form complete (modular) lattices.

Projective Geometry on Modular Lattices

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