Convex Analysis for Optimization by Jan Brinkhuis
Author:Jan Brinkhuis
Language: eng
Format: epub
ISBN: 9783030418045
Publisher: Springer International Publishing
2.The image and inverse image of a polyhedral set under a linear transformation are again polyhedral sets.
3.The polyhedral property is preserved by the standard binary operations for convex sets—sum, intersection, convex hull of the union, inverse sum.
4.For polyhedral sets containing the origin, the following formulas hold
5.For polyhedral sets containing the origin for which and , the following formula holds:
We do not go into the definitions of vertices, edges and facets of polyhedral sets. However we invite you to think about what happens to the structure of vertices, edges and facets of a polyhedral set when you apply the polar set operator. Try to guess this. Hint: try to figure out first the case of the platonic solids—tetrahedron, cube, octahedron, dodecahedron, icosahedron—if they contain the origin.
Conclusion
Polyhedral sets containing the origin are the nicest class of convex sets. In particular, the polyhedral set property is preserved under taking the image under a linear transformation and under taking the polar set. This makes the duality theory for polyhedral sets very simple: it involves no closure operator.
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Algebraic Geometry | Analytic Geometry |
Differential Geometry | Non-Euclidean Geometries |
Topology |
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