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. ${\displaystyle M}$ ${\displaystyle n}$ ${\displaystyle V}$ ${\displaystyle \operatorname {aff} (M)}$ ${\displaystyle M}$ ${\displaystyle V}$ ${\displaystyle x}$ ${\displaystyle M}$ ${\displaystyle M}$ ${\displaystyle U_{\epsilon }(x)}$ ${\displaystyle x}$ ${\displaystyle U_{\epsilon }(x)\cap \operatorname {aff} (M)\subset M}$ ${\displaystyle M}$ ${\displaystyle \operatorname {aff} (M)}$ ${\displaystyle M}$ ${\displaystyle \operatorname {relint} (M)}$

Difference between the inner point and the relatively inner point of a set

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 . ${\displaystyle C}$ ${\displaystyle C}$ ${\displaystyle f\colon I\rightarrow \mathbb {R} ^{2}}$ ${\displaystyle I\subseteq \mathbb {R} }$

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. ${\displaystyle f(t)}$ ${\displaystyle t}$ ${\displaystyle I}$ ${\displaystyle t}$ ${\displaystyle f}$ ${\displaystyle t}$