Relatively internal point
The term relative internal point is a topological term that is used in mathematical optimization .
definition
Let be a subset of a -dimensional real vector space , the affine hull of in . Then a point from is called a relatively inner point from if there is a neighborhood of such that it holds. The relatively inner points of are therefore exactly the inner points with respect to the subspace topology of . The set of all relatively interior points is called the relatively interior of the set and is denoted by.
Examples
Cuboid
We consider a cuboid in three-dimensional (real) space. Then:
- A point inside the cuboid is a relatively internal point of the solid cuboid.
- A point on a side surface of the cuboid (not on an edge) is a relatively internal point of the relevant side surface, but not the solid cuboid.
- A point on an edge of the cuboid that is not a corner point of the cuboid is a relatively inner point of the relevant edge, but neither a side surface nor the solid cuboid.
- A corner point of the cuboid is not a relatively internal point in any (nontrivial) subset of the cuboid.
Circular disc
We consider a closed circular disk in three-dimensional (real) space. Then:
- The affine envelope of the circular disk is the plane in space in which the circle lies.
- The points of the circular line are not relatively internal points for the circular disk.
- All other points on the circular disk are relatively internal points.
Curve in the plane
Be a curve in the plane. Formal: be the image of a continuous function on an interval .
A point on the curve that is neither its start nor its end point (that is, lies in the interior of ) is a relatively inner point of the curve if and only if the curve goes straight ahead in a neighborhood of . If the function is differentiable twice at this point, this means that the curve has the curvature 0 there.