Amelunxen, Dennis: Geometric analysis of the condition of the convex feasibility problem. 2011
Inhalt
- Introduction
- Complexity of numerical algorithms
- The convex feasibility problem
- Tube formulas
- Results
- Outline
- Credits
- The Grassmann condition
- Preliminaries from numerical linear algebra
- The homogeneous convex feasibility problem
- Defining the Grassmann condition
- Spherical convex geometry
- Some basic definitions
- The metric space of spherical convex sets
- Subfamilies of spherical convex sets
- Spherical tube formulas
- Preliminaries
- The Weingarten map for submanifolds of Rn
- Smooth caps
- Integration on submanifolds of Rn
- The binomial coefficient and related quantities
- The (euclidean) Steiner polynomial
- Weyl's tube formulas
- Spherical intrinsic volumes
- Computations in the Grassmann manifold
- Preliminary: Riemannian manifolds
- Orthogonal group
- Quotients of the orthogonal group
- Geodesics in Gr(n,m)
- Closest elements in the Sigma set
- A tube formula for the Grassmann bundle
- Main results
- Parametrizing the Sigma set
- Computing the tube
- The expected twisted characteristic polynomial
- Proof of Theorem 6.1.1
- Estimations
- Miscellaneous
- Some computation rules for intrinsic volumes
- The semidefinite cone
- Preliminary: Some integrals appearing
- The intrinsic volumes of the semidefinite cone
- Observations, open questions, conjectures
- On the distribution of the principal angles
- Bibliography