skip to content
Convex bodies : the Brunn-Minkowski theory Preview this item
ClosePreview this item
Checking...

Convex bodies : the Brunn-Minkowski theory

Author: Rolf Schneider
Publisher: Cambridge ; New York : Cambridge University Press, ©1993.
Series: Encyclopedia of mathematics and its applications, v. 44.
Edition/Format:   Print book : EnglishView all editions and formats
Summary:
"At the heart of this monograph is the Brunn-Minkowski theory. It can be used to great effect in studying such ideas as volume and surface area and the generalizations of these. In particular the notions of mixed volume and mixed area arise naturally and the fundamental inequalities that are satisfied by mixed volumes are considered in detail." "The author presents a comprehensive introduction to convex bodies and  Read more...
Rating:

(not yet rated) 0 with reviews - Be the first.

Subjects
More like this

 

Find a copy online

Links to this item

Find a copy in the library

&AllPage.SpinnerRetrieving; Finding libraries that hold this item...

Details

Material Type: Internet resource
Document Type: Book, Internet Resource
All Authors / Contributors: Rolf Schneider
ISBN: 0521352207 9780521352208 9780521059916 0521059917
OCLC Number: 25629815
Description: xiii, 490 pages : illustrations ; 24 cm.
Contents: 1. Basic convexity. 1.1. Convex sets and combinations. 1.2. The metric projection. 1.3. Support and separation. 1.4. Extremal representations. 1.5. Convex functions. 1.6. Duality. 1.7. The support function. 1.8. The Hausdorff metric --
2. Boundary structure. 2.1. Facial structure. 2.2. Singularities. 2.3. Segments in the boundary. 2.4. Polytopes. 2.5. Higher regularity and curvature. 2.6. Generic boundary structure --
3. Minkowski addition. 3.1. Minkowski addition and subtraction. 3.2. Summands and decomposition. 3.3. Approximation and addition. 3.4. Additive maps. 3.5. Zonoids and other classes of convex bodies --
4. Curvature measures and quermassintegrals. 4.1. Local parallel sets. 4.2. Curvature measures and area measures. 4.3. The area measure of order one. 4.4. Additive extension. 4.5. Integral-geometric formulae. 4.6. Local behaviour of curvature measures --
5. Mixed volumes and related concepts. 5.1. Mixed volumes and mixed area measures. 5.2. Extensions of mixed volumes. 5.3. Special formulae for mixed volumes and quermassintegrals. 5.4. Moment vectors and curvature centroids --
6. Inequalities for mixed volumes. 6.1. The Brunn-Minkowski theorem. 6.2. The Minkowski and isoperimetric inequalities. 6.3. The Aleksandrov-Fenchel inequality. 6.4. Consequences and improvements. 6.5. Generalized parallel bodies. 6.6. Equality cases and stability. 6.7. Linear inequalities. 6.8. Analogous notions and inequalities --
7. Selected applications. 7.1. Minkowski's existence theorem. 7.2. Uniqueness theorems for area measures. 7.3. The difference-body inequality. 7.4. Affinely associated bodies --
Appendix: Spherical harmonics.
Series Title: Encyclopedia of mathematics and its applications, v. 44.
Responsibility: Rolf Schneider.
More information:

Abstract:

A comprehensive introduction to convex bodies giving full proofs for some deeper theorems which have never previously been brought together.  Read more...

Reviews

Editorial reviews

Publisher Synopsis

"A replacement for Bonnesen and Fenchel's text has been very much needed; the monograph under review provides a most satisfactory one." W.J. Firey, Mathematical Reviews

 
User-contributed reviews
Retrieving GoodReads reviews...
Retrieving DOGObooks reviews...

Tags

Be the first.
Confirm this request

You may have already requested this item. Please select Ok if you would like to proceed with this request anyway.

Linked Data


Primary Entity

<http://www.worldcat.org/oclc/25629815> # Convex bodies : the Brunn-Minkowski theory
    a schema:CreativeWork, schema:Book ;
   library:oclcnum "25629815" ;
   library:placeOfPublication <http://id.loc.gov/vocabulary/countries/enk> ;
   library:placeOfPublication <http://dbpedia.org/resource/New_York_City> ; # New York
   library:placeOfPublication <http://experiment.worldcat.org/entity/work/data/806673690#Place/cambridge> ; # Cambridge
   schema:about <http://experiment.worldcat.org/entity/work/data/806673690#Topic/corps_convexes> ; # Corps convexes
   schema:about <http://experiment.worldcat.org/entity/work/data/806673690#Thing/analytic_geometry> ; # Analytic geometry
   schema:about <http://id.worldcat.org/fast/877258> ; # Convex bodies
   schema:about <http://experiment.worldcat.org/entity/work/data/806673690#Topic/konvexer_korper> ; # Konvexer Körper
   schema:about <http://experiment.worldcat.org/entity/work/data/806673690#Topic/geometrie_analytique> ; # Géométrie analytique
   schema:about <http://dewey.info/class/516.374/e20/> ;
   schema:bookFormat bgn:PrintBook ;
   schema:copyrightYear "1993" ;
   schema:creator <http://viaf.org/viaf/64074069> ; # Rolf Schneider
   schema:datePublished "1993" ;
   schema:description "1. Basic convexity. 1.1. Convex sets and combinations. 1.2. The metric projection. 1.3. Support and separation. 1.4. Extremal representations. 1.5. Convex functions. 1.6. Duality. 1.7. The support function. 1.8. The Hausdorff metric -- 2. Boundary structure. 2.1. Facial structure. 2.2. Singularities. 2.3. Segments in the boundary. 2.4. Polytopes. 2.5. Higher regularity and curvature. 2.6. Generic boundary structure -- 3. Minkowski addition. 3.1. Minkowski addition and subtraction. 3.2. Summands and decomposition. 3.3. Approximation and addition. 3.4. Additive maps. 3.5. Zonoids and other classes of convex bodies -- 4. Curvature measures and quermassintegrals. 4.1. Local parallel sets. 4.2. Curvature measures and area measures. 4.3. The area measure of order one. 4.4. Additive extension. 4.5. Integral-geometric formulae. 4.6. Local behaviour of curvature measures -- 5. Mixed volumes and related concepts. 5.1. Mixed volumes and mixed area measures. 5.2. Extensions of mixed volumes. 5.3. Special formulae for mixed volumes and quermassintegrals. 5.4. Moment vectors and curvature centroids -- 6. Inequalities for mixed volumes. 6.1. The Brunn-Minkowski theorem. 6.2. The Minkowski and isoperimetric inequalities. 6.3. The Aleksandrov-Fenchel inequality. 6.4. Consequences and improvements. 6.5. Generalized parallel bodies. 6.6. Equality cases and stability. 6.7. Linear inequalities. 6.8. Analogous notions and inequalities -- 7. Selected applications. 7.1. Minkowski's existence theorem. 7.2. Uniqueness theorems for area measures. 7.3. The difference-body inequality. 7.4. Affinely associated bodies -- Appendix: Spherical harmonics."@en ;
   schema:exampleOfWork <http://worldcat.org/entity/work/id/806673690> ;
   schema:inLanguage "en" ;
   schema:isPartOf <http://experiment.worldcat.org/entity/work/data/806673690#Series/encyclopedia_of_mathematics_and_its_applications> ; # Encyclopedia of mathematics and its applications ;
   schema:name "Convex bodies : the Brunn-Minkowski theory"@en ;
   schema:productID "25629815" ;
   schema:publication <http://www.worldcat.org/title/-/oclc/25629815#PublicationEvent/cambridge_new_york_cambridge_university_press_1993> ;
   schema:publisher <http://experiment.worldcat.org/entity/work/data/806673690#Agent/cambridge_university_press> ; # Cambridge University Press
   schema:reviews <http://www.worldcat.org/title/-/oclc/25629815#Review/1486581969> ;
   schema:url <http://catdir.loc.gov/catdir/toc/cam027/92011481.html> ;
   schema:url <http://catdir.loc.gov/catdir/samples/cam034/92011481.html> ;
   schema:url <http://www.gbv.de/dms/ilmenau/toc/11228454X.PDF> ;
   schema:workExample <http://worldcat.org/isbn/9780521059916> ;
   schema:workExample <http://worldcat.org/isbn/9780521352208> ;
   umbel:isLike <http://bnb.data.bl.uk/id/resource/GB9313461> ;
   wdrs:describedby <http://www.worldcat.org/title/-/oclc/25629815> ;
    .


Related Entities

<http://dbpedia.org/resource/New_York_City> # New York
    a schema:Place ;
   schema:name "New York" ;
    .

<http://experiment.worldcat.org/entity/work/data/806673690#Agent/cambridge_university_press> # Cambridge University Press
    a bgn:Agent ;
   schema:name "Cambridge University Press" ;
    .

<http://experiment.worldcat.org/entity/work/data/806673690#Series/encyclopedia_of_mathematics_and_its_applications> # Encyclopedia of mathematics and its applications ;
    a bgn:PublicationSeries ;
   schema:hasPart <http://www.worldcat.org/oclc/25629815> ; # Convex bodies : the Brunn-Minkowski theory
   schema:name "Encyclopedia of mathematics and its applications ;" ;
    .

<http://experiment.worldcat.org/entity/work/data/806673690#Thing/analytic_geometry> # Analytic geometry
    a schema:Thing ;
   schema:name "Analytic geometry" ;
    .

<http://experiment.worldcat.org/entity/work/data/806673690#Topic/corps_convexes> # Corps convexes
    a schema:Intangible ;
   schema:name "Corps convexes"@en ;
   schema:name "Corps convexes"@fr ;
    .

<http://experiment.worldcat.org/entity/work/data/806673690#Topic/geometrie_analytique> # Géométrie analytique
    a schema:Intangible ;
   schema:name "Géométrie analytique"@en ;
    .

<http://id.worldcat.org/fast/877258> # Convex bodies
    a schema:Intangible ;
   schema:name "Convex bodies"@en ;
    .

<http://viaf.org/viaf/64074069> # Rolf Schneider
    a schema:Person ;
   schema:birthDate "1940" ;
   schema:familyName "Schneider" ;
   schema:givenName "Rolf" ;
   schema:name "Rolf Schneider" ;
    .

<http://worldcat.org/isbn/9780521059916>
    a schema:ProductModel ;
   schema:isbn "0521059917" ;
   schema:isbn "9780521059916" ;
    .

<http://worldcat.org/isbn/9780521352208>
    a schema:ProductModel ;
   schema:isbn "0521352207" ;
   schema:isbn "9780521352208" ;
    .

<http://www.worldcat.org/title/-/oclc/25629815#Review/1486581969>
    a schema:Review ;
   schema:itemReviewed <http://www.worldcat.org/oclc/25629815> ; # Convex bodies : the Brunn-Minkowski theory
   schema:reviewBody ""At the heart of this monograph is the Brunn-Minkowski theory. It can be used to great effect in studying such ideas as volume and surface area and the generalizations of these. In particular the notions of mixed volume and mixed area arise naturally and the fundamental inequalities that are satisfied by mixed volumes are considered in detail." "The author presents a comprehensive introduction to convex bodies and gives full proofs for some deeper theorems which have never previously been brought together. Many hints and pointers to connections with other fields are given, and an exhaustive reference list is included."--Jacket." ;
    .


Content-negotiable representations

Close Window

Please sign in to WorldCat 

Don't have an account? You can easily create a free account.