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Let a subset of
. We say that is convex when, for any pair of points in , the segment lies entirely inside .
The former statement is equivalent to saying that for any pair of vectors in , the vector is in for all .
If is a convex set, for any
in , and any positive numbers
such that
the vector
is in .
Examples of convex sets in the plane are circles, triangles, and ellipses. The definition given above can be generalized to any real vector space:
Let be a vector space (over or ). A subset of is convex if for all points in , the line segment
is also in .
More generally, the same definition works for any vector space over an ordered field.
A polyconvex set is a finite union of compact, convex sets.
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"convex set" is owned by drini. [ full author list (3) | owner history (1) ]
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(view preamble)
Cross-references: compact, union, finite, ordered field, line segment, vector space, real, ellipses, triangles, circles, plane, positive, vectors, equivalent, segment, points, subset
There are 94 references to this entry.
This is version 14 of convex set, born on 2001-10-15, modified 2007-06-18.
Object id is 243, canonical name is ConvexSet.
Accessed 20032 times total.
Classification:
| AMS MSC: | 52A99 (Convex and discrete geometry :: General convexity :: Miscellaneous) |
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Pending Errata and Addenda
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