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.

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Convex bodies : the Brunn-Minkowski theory

Metric Spaces Length Spaces Constructions Spaces of Bounded Curvature Smooth Length Structures Curvature of Riemannian Metrics Space of Metric Spaces Large-scale Geometry Spaces of Curvature Bounded Above Spaces of Curvature Bounded Below Bibliography Index.

/pdf/a-course-in-metric-geometry-kree1qe21g.pdf

A Course in Metric Geometry

Summary. The 
$L^2$
 Monge-Kantorovich mass transfer problem [31] is reset in a fluid mechanics framework and numerically solved by an augmented Lagrangian method.

A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem

Abstract: A new method is proposed for multi-objective optimization with discrete alternatives: MOORA (Multi-Objective Optimization on the basis of Ratio Analysis). This method refers to a matrix of responses of alternatives to objectives, to which ratios are applied. A well established other method for multi-objective optimization is used for comparison, namely the reference point method. Later on, it is demonstrated that this is the best choice among the different competing methods. In MOORA the set of ratios has the square roots of the sum of squared responses as denominators. These ratios, as dimensionless, seem to be the best choice among different ratios. These dimensionless ratios, situated between zero and one, are added in the case of maximization or subtracted in case of minimization. Finally, all alternatives are ranked, according to the obtained ratios. Eventually, to give more importance to an objective, an objective can be replaced by different sub-objectives or a coefficient of importance can be specified. An example on privatization in a transition economy illustrates the application of the method. If application is situated originally in a “welfare” economy, centered on production, MOORA becomes even more significant in a “wellbeing economy”, where consumer sovereignty is assumed.

The MOORA method and its application to privatization in a transition economy

The Brunn-Minkowski theory is the heart of quantitative convexity. It had its origins in Minkowski's joining his notion of mixed volumes with the Brunn-Minkowski inequality. One of Minkowski's major contributions to the theory was to show how this theory could be developed from a few basic concepts: support functions, Minkowski combinations, and mixed volumes. Thirty years ago, Firey [8] (see Burago and Zalgaller [4, §24.6]) extended the notion of a Minkowski combination, and introduced, for each real p > 1, what he called p-sums. It is the aim of this series of articles to show that these Firey combinations lead to a Brunn-Minkowski theory for each p > 1. Let Jf denote the set of convex bodies (compact, convex subsets with nonempty interiors) in Euclidean λz-space, R . Let 3£\" denote the set of convex bodies containing the origin in their interiors. For K e 3? , let hκ = h(K, •): S ~ -> R denote the support function of K\\ i.e., for u e S~, hκ{u) = h(K, ύ) = max{w x : x e K} , where u-x denotes the standard inner product in R . The set Jf will be viewed as equipped with the usual Hausdorff metric, d, defined by d(K, L) = \\hκ AJ^ , where | 1^ is the sup (or max) norm on the space of continuous functions on the unit sphere, C(S~). For K, L e J£ , and α, β > 0 (not both zero), the Minkowski linear combination aK + βL e J? is defined by

/pdf/the-brunn-minkowski-firey-theory-i-mixed-volumes-and-the-4pls2rcl95.pdf

The Brunn-Minkowski-Firey theory. I. Mixed volumes and the Minkowski problem

If you really want to be smarter, reading can be one of the lots ways to evoke and realize. Many people who like reading will have more knowledge and experiences. Reading can be a way to gain information from economics, politics, science, fiction, literature, religion, and many others. As one of the part of book categories, intrinsic geometry of convex surfaces always becomes the most wanted book. Many people are absolutely searching for this book. It means that many love to read this kind of book.

Extrinsic geometry of convex surfaces

The Minkowski multidimensional problem

On the improper convex affine hyperspheres

Bendings of surfaces and elastic deformations of shells Elastic states of general convex shells Supercritical elastic states of a cylindrical shell under axial compression Infinitesimal bendings of surfaces and the stability of shells Loss of stability of a cylindrical shell under axial compression.

A. V. Pogorelov

Papers

Extrinsic geometry of convex surfaces

The Minkowski multidimensional problem

On the improper convex affine hyperspheres

Bendings of surfaces and stability of shells

Monge-Ampère equations of elliptic type