Content deleted Content added
split paragraph in two, some wording adjustments Tags: Mobile edit Mobile app edit iOS app edit |
m italics—>bold for terms Tags: Mobile edit Mobile app edit iOS app edit |
||
Line 18:
This means that the [[affine combination]] {{math|(1 − ''t'')''x'' + ''ty''}} belongs to {{mvar|C}} for all {{mvar|x,y}} in {{mvar|C}} and {{mvar|t}} in the [[interval (mathematics)|interval]] {{math|[0, 1]}}. This implies that convexity is invariant under [[affine transformation]]s. Further, it implies that a convex set in a [[real number|real]] or [[complex number|complex]] [[topological vector space]] is [[path-connected]], thus [[connected space|connected]].
A set {{mvar|C}} is '''{{visible anchor|strictly convex}}''' if every point on the line segment connecting {{mvar|x}} and {{mvar|y}} other than the endpoints is inside the [[Interior (topology)|topological interior]] of {{mvar|C}}. A closed convex subset is strictly convex if and only if every one of its [[Boundary (topology)|boundary points]] is an [[extreme point]].<ref>{{Halmos A Hilbert Space Problem Book 1982|p=5}}</ref>
A set {{mvar|C}} is '''[[absolutely convex]]''' if it is convex and [[balanced set|balanced]].
The convex [[subset]]s of {{math|'''R'''}} (the set of real numbers) are the intervals and the points of {{math|'''R'''}}. Some examples of convex subsets of the [[Euclidean plane]] are solid [[regular polygon]]s, solid triangles, and intersections of solid triangles. Some examples of convex subsets of a [[Euclidean space|Euclidean 3-dimensional space]] are the [[Archimedean solid]]s and the [[Platonic solid]]s. The [[Kepler-Poinsot polyhedra]] are examples of non-convex sets.
| |||