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- Illinois Journal of Mathematics
- Pairs of Compact Convex Sets
- Pairs of Compact Convex Sets - Diethard Pallaschke - Innbundet () » Bokkilden
- Convex hull.
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For each choice of coefficients, the resulting convex combination is a point in the convex hull, and the whole convex hull can be formed by choosing coefficients in all possible ways. Expressing this as a single formula, the convex hull is the set:. In two dimensions, the convex hull is sometimes partitioned into two polygonal chains, the upper hull and the lower hull, stretching between the leftmost and rightmost points of the hull.
Illinois Journal of Mathematics
More generally, for points in any dimension in general position, each facet of the convex hull is either oriented upwards separating the hull from points directly above it or downwards; the union of the upward-facing facets forms a topological disk, the upper hull, and similarly the union of the downward-facing facets forms the lower hull. In computational geometry, a number of algorithms are known for computing the convex hull for a finite set of points and for other geometric objects.
Computing the convex hull means constructing an unambiguous, efficient representation of the required convex shape. The complexity of the corresponding algorithms is usually estimated in terms of n , the number of input points, and h , the number of points on the convex hull. The operation of taking convex hulls behaves well with respect to the Minkowski addition of sets. The convex-hull operation is needed for the set of convex sets to form a lattice , in which the "join" operation is the convex hull of the union of two convex sets:.
The Delaunay triangulation of a point set and its dual , the Voronoi diagram , are mathematically related to convex hulls: Topologically, the convex hull of an open set is always itself open, and the convex hull of a compact set is always itself compact; however, there exist closed sets for which the convex hull is not closed. The problem of finding convex hulls finds its practical applications in pattern recognition , image processing , statistics , geographic information system , game theory , construction of phase diagrams , and static code analysis by abstract interpretation.
It also serves as a tool, a building block for a number of other computational-geometric algorithms such as the rotating calipers method for computing the width and diameter of a point set.
The convex hull is commonly known as the minimum convex polygon MCP in ethology , where it is a classic, though perhaps simplistic, approach in estimating an animal's home range based on points where the animal has been observed. From Wikipedia, the free encyclopedia.
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Convex hull calculated by SciPy [3] left for the coordinates of the truncated cuboctahedron right , containing 92 triangular simplices some of them coplanar. Minkowski addition and Shapley—Folkman lemma. The idea of partitioning the hull into two chains comes from an efficient variant of Graham scan by Andrew Riesz extension Riesz representation Open mapping Parseval's identity Schauder fixed-point.
Pairs of Compact Convex Sets
Retrieved from " https: Separate different tags with a comma. To include a comma in your tag, surround the tag with double quotes. Skip to content Skip to search. Pairs of compact convex sets: Published Dordrecht ; Boston: Kluwer Academic Publishers, c Language English View all editions Prev Next edition 2 of 3. Check copyright status Cite this Title Pairs of compact convex sets: Physical Description xii, p.
Pairs of Compact Convex Sets - Diethard Pallaschke - Innbundet () » Bokkilden
Series Mathematics and its applications ; v. Convex sets and sublinearity 2. Topological vector spaces 3.
Compact convex sets 4. Minimal pairs of convex sets 5. The cardinality of minimal pairs 6. Minimality under constraints 7. Notes Includes bibliographical references p. View online Borrow Buy Freely available Show 0 more links Related resource Table of contents only at http: Set up My libraries How do I set up "My libraries"?