mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, the relative interior of a set is a refinement of the concept of the

interior Interior may refer to: Arts and media * ''Interior'' (Degas) (also known as ''The Rape''), painting by Edgar Degas * ''Interior'' (play), 1895 play by Belgian playwright Maurice Maeterlinck * ''The Interior'' (novel), by Lisa See * Interior de ...

, which is often more useful when dealing with low-dimensional sets placed in higher-dimensional spaces. Formally, the relative interior of a set

S

(denoted

\operatorname(S)

) is defined as its

within the affine hull of

S.

In other words,

\operatorname(S) := \,

where

\operatorname(S)

is the affine hull of

S,

and

N_\epsilon(x)

is a

ball A ball is a round object (usually spherical, but can sometimes be ovoid) with several uses. It is used in ball games, where the play of the game follows the state of the ball as it is hit, kicked or thrown by players. Balls can also be used f ...

of radius

\epsilon

centered on

x

. Any metric can be used for the construction of the ball; all metrics define the same set as the relative interior. For any nonempty convex set

C \subseteq \mathbb^n

the relative interior can be defined as

\operatorname(C) := \.

Comparison to interior

* The interior of a point is empty, but its relative interior is the point itself. * The interior of a line segment on the 2D plane is empty, but its relative interior is the line segment without its endpoints. * The interior of a disc in 3D space is empty, but its relative interior is the same disc without its circular edge.

Comparison to interior

See also

References

Further reading