Relative interior

The term relative Inner point is a topological term that is used in mathematical optimization.

Definition

Let M be a subset of an n- dimensional real vector space V, aff (M) the affine hull of M in V. Then is, a point x of M relative interior point of M, if it is in the affine space aff (M), provided with the subspace topology, a neighborhood of x there that is completely contained in M.

Examples

Cuboid

We consider a cube in three-dimensional (real ) space. Then:

  • A point in the interior of the box is relatively inner point of the solid parallelepiped.
  • A point on a side surface of the box ( not on an edge ) is relatively inner point of the relevant side surface, but not the full- cube.
  • A point on an edge of the parallelepiped, other than a corner of the box is relatively inner point of the respective edge, but not a side face of the full or cuboid.
  • A corner of the box is a relative interior point in any ( non-trivial ) subset of the box.

Circular disk

We consider a closed disk in three-dimensional (real ) space. Then:

  • The affine case of the circular disc is the plane in space, in which the circle is.
  • The points of the circle are not relative interior points of the circular disk.
  • All other points of the circular disk are relatively interior points.

Curve in the plane

Be a curve in the plane. Formal: is the image of a continuous function on an interval.

A point f (t) on the curve, it is neither the beginning nor the end point it is (that is, t is in the inside of I) if and only a relatively inner point of the curve if the curve is in an area of t ahead. Is if the function f at the position t twice differentiable, this means that the curve where the curvature is 0).

  • Set topology
677413
de