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N-dimensional Polyhedron
An ''n''-dimensional polyhedron is a geometric object that generalizes the 3-dimensional polyhedron to an ''n''-dimensional space. It is defined as a set of points in real affine (or Euclidean) space of any dimension ''n'', that has flat sides. It may alternatively be defined as the intersection of finitely many half-spaces. Unlike a 3-dimensional polyhedron, it may be bounded or unbounded. In this terminology, a bounded polyhedron is called a polytope... Analytically, a convex polyhedron is expressed as the solution set for a system of linear inequalities, ''ai''T''x'' ≤ ''bi'', where ''ai'' are vectors in R''n'' and ''bi'' are scalars. This definition of polyhedra is particularly important as it provides a geometric perspective for problems in linear programming. Examples Many traditional polyhedral forms are n-dimensional polyhedra. Other examples include: * A half-space is a polyhedron defined by a single linear inequality, ''a1''T''x'' ≤ ''b1''. * A hyperplane is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polyhedron
In geometry, a polyhedron (: polyhedra or polyhedrons; ) is a three-dimensional figure with flat polygonal Face (geometry), faces, straight Edge (geometry), edges and sharp corners or Vertex (geometry), vertices. The term "polyhedron" may refer either to a solid figure or to its boundary surface (mathematics), surface. The terms solid polyhedron and polyhedral surface are commonly used to distinguish the two concepts. Also, the term ''polyhedron'' is often used to refer implicitly to the whole structure (mathematics), structure formed by a solid polyhedron, its polyhedral surface, its faces, its edges, and its vertices. There are many definitions of polyhedron. Nevertheless, the polyhedron is typically understood as a generalization of a two-dimensional polygon and a three-dimensional specialization of a polytope, a more general concept in any number of dimensions. Polyhedra have several general characteristics that include the number of faces, topological classification by Eule ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Boundary (topology)
In topology and mathematics in general, the boundary of a subset of a topological space is the set of points in the Closure (topology), closure of not belonging to the Interior (topology), interior of . An element of the boundary of is called a boundary point of . The term boundary operation refers to finding or taking the boundary of a set. Notations used for boundary of a set include \operatorname(S), \operatorname(S), and \partial S. Some authors (for example Willard, in ''General Topology'') use the term frontier instead of boundary in an attempt to avoid confusion with a Manifold#Manifold with boundary, different definition used in algebraic topology and the theory of manifolds. Despite widespread acceptance of the meaning of the terms boundary and frontier, they have sometimes been used to refer to other sets. For example, ''Metric Spaces'' by E. T. Copson uses the term boundary to refer to Felix Hausdorff, Hausdorff's border, which is defined as the intersection ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algorithmic Problems On Convex Sets
Many problems in mathematical programming can be formulated as problems on convex sets or convex bodies. Six kinds of problems are particularly important: optimization, violation, validity, separation, membership and emptiness. Each of these problems has a strong (exact) variant, and a weak (approximate) variant. In all problem descriptions, ''K'' denotes a compact and convex set in R''n''. Strong variants The strong variants of the problems are: * Strong optimization problem (SOPT): given a vector ''c'' in R''n'', find a vector ''y'' in ''K'' such that ''c''T''y'' ≥ ''c''T''x'' for all ''x'' in ''K'', or assert that ''K'' is empty. * Strong violation problem (SVIOL): given a vector ''c'' in R''n'' and a number ''t'', decide whether ''c''T''x'' ≤ ''t'' for all ''x'' in ''K'', or find ''y'' in ''K'' such that ''c''T''y'' > ''t''. * Strong validity problem (SVAL): given a vector ''c'' in R''n'' and a number ''t'', decide whether ''c''T''x'' ≤ ''t'' for all ''x'' in ''K''. * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
