Extreme Set
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In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, most commonly in
convex geometry In mathematics, convex geometry is the branch of geometry studying convex sets, mainly in Euclidean space. Convex sets occur naturally in many areas: computational geometry, convex analysis, discrete geometry, functional analysis, geometry of num ...
, an extreme set or face of a set C\subseteq V in a
vector space In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...
V is a subset F\subseteq C with the property that if for any two points x,y\in C some in-between point z=\theta x + (1-\theta) y,\theta\in ,1/math> lies in F, then we must have had x,y\in F. An
extreme point In mathematics, an extreme point of a convex set S in a Real number, real or Complex number, complex vector space is a point in S that does not lie in any open line segment joining two points of S. The extreme points of a line segment are calle ...
of C is a point p\in C for which \ is a face. An
exposed face In mathematics, most commonly in convex geometry, an extreme set or face of a set C\subseteq V in a vector space V is a subset F\subseteq C with the property that if for any two points x,y\in C some in-between point z=\theta x + (1-\theta) y,\the ...
of C is the subset of points of C where a linear functional achieves its minimum on C. Thus, if f is a linear functional on V and \alpha =\inf\>-\infty, then \ is an exposed face of C. An
exposed point In mathematics, an exposed point of a convex set C is a point x\in C at which some continuous linear functional attains its strict maximum over C. Such a functional is then said to ''expose'' x. There can be many exposing functionals for x. The s ...
of C is a point p\in C such that \ is an exposed face. That is, f(p) > f(c) for all c\in C\setminus\. An exposed face is a face, but the converse is not true (see the figure). An exposed face of C is convex if C is convex. If F is a face of C\subseteq V, then E\subseteq F is a face of F if and only if E is a face of C.


Competing definitions

Some authors do not include C and/or \varnothing among the (exposed) faces. Some authors require F and/or C to be
convex Convex or convexity may refer to: Science and technology * Convex lens, in optics Mathematics * Convex set, containing the whole line segment that joins points ** Convex polygon, a polygon which encloses a convex set of points ** Convex polytop ...
(else the boundary of a disc is a face of the disc, as well as any subset of the boundary) or closed. Some authors require the functional f to be continuous in a given
vector topology In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is als ...
.


See also

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Face (geometry) In solid geometry, a face is a flat surface (a Plane (geometry), planar region (mathematics), region) that forms part of the boundary of a solid object. For example, a cube has six faces in this sense. In more modern treatments of the geometry of ...


References


Bibliography

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External links


TOPOLOGICAL VECTOR SPACES AND CONTINUOUS LINEAR FUNCTIONALS
Chapter III of FUNCTIONAL ANALYSIS, Lawrence Baggett, University of Colorado Boulder.
Functional Analysis
Peter Philip, Ludwig-Maximilians-universität München, 2024 Convex geometry Convex hulls Functional analysis Mathematical analysis