Assume that ${\displaystyle A}$  is a subset of a vector space ${\displaystyle X.}$  The algebraic interior (or radial kernel) of ${\displaystyle A}$  with respect to ${\displaystyle X}$  is the set of all points at which ${\displaystyle A}$  is a radial set. A point ${\displaystyle a_{0}\in A}$  is called an internal point of ${\displaystyle A}$ ^[1][2] and ${\displaystyle A}$  is said to be radial at ${\displaystyle a_{0}}$  if for every ${\displaystyle x\in X}$  there exists a real number ${\displaystyle t_{x}>0}$  such that for every ${\displaystyle t\in [0,t_{x}],}$  ${\displaystyle a_{0}+tx\in A.}$  This last condition can also be written as ${\displaystyle a_{0}+[0,t_{x}]x\subseteq A}$  where the set

If ${\displaystyle M}$  is a linear subspace of ${\displaystyle X}$  and ${\displaystyle A\subseteq X}$  then this definition can be generalized to the algebraic interior of ${\displaystyle A}$  with respect to ${\displaystyle M}$  is:^[4]

Algebraic closure

A point ${\displaystyle x\in X}$  is said to be linearly accessible from a subset ${\displaystyle A\subseteq X}$  if there exists some ${\displaystyle a\in A}$  such that the line segment ${\displaystyle [a,x):=a+[0,1)x}$  is contained in ${\displaystyle A.}$ ^[5] The algebraic closure of ${\displaystyle A}$  with respect to ${\displaystyle X}$ , denoted by ${\displaystyle \operatorname {acl} _{X}A,}$  consists of ${\displaystyle A}$  and all points in ${\displaystyle X}$  that are linearly accessible from ${\displaystyle A.}$ ^[5]

In the special case where ${\displaystyle M:=X,}$  the set ${\displaystyle \operatorname {aint} _{X}A}$  is called the algebraic interior or core of ${\displaystyle A}$  and it is denoted by ${\displaystyle A^{i}}$  or ${\displaystyle \operatorname {core} A.}$  Formally, if ${\displaystyle X}$  is a vector space then the algebraic interior of ${\displaystyle A\subseteq X}$  is^[6]

If ${\displaystyle A}$  is non-empty, then these additional subsets are also useful for the statements of many theorems in convex functional analysis (such as the Ursescu theorem):

If ${\displaystyle X}$  is a Fréchet space, ${\displaystyle A}$  is convex, and ${\displaystyle \operatorname {aff} A}$  is closed in ${\displaystyle X}$  then ${\displaystyle {}^{ic}A={}^{ib}A}$  but in general it is possible to have ${\displaystyle {}^{ic}A=\varnothing }$  while ${\displaystyle {}^{ib}A}$  is not empty.

Examples edit

If ${\displaystyle A=\{x\in \mathbb {R} ^{2}:x_{2}\geq x_{1}^{2}{\text{ or }}x_{2}\leq 0\}\subseteq \mathbb {R} ^{2}}$  then ${\displaystyle 0\in \operatorname {core} (A),}$  but ${\displaystyle 0\not \in \operatorname {int} (A)}$  and ${\displaystyle 0\not \in \operatorname {core} (\operatorname {core} (A)).}$ 

Properties of core edit

Suppose ${\displaystyle A,B\subseteq X.}$ 

• In general, ${\displaystyle \operatorname {core} A\neq \operatorname {core} (\operatorname {core} A).}$  But if ${\displaystyle A}$  is a convex set then:
  • ${\displaystyle \operatorname {core} A=\operatorname {core} (\operatorname {core} A),}$  and
  • for all ${\displaystyle x_{0}\in \operatorname {core} A,y\in A,0<\lambda \leq 1}$  then ${\displaystyle \lambda x_{0}+(1-\lambda )y\in \operatorname {core} A.}$ 
• ${\displaystyle A}$  is an absorbing subset of a real vector space if and only if ${\displaystyle 0\in \operatorname {core} (A).}$ ^[3]
• ${\displaystyle A+\operatorname {core} B\subseteq \operatorname {core} (A+B)}$ ^[7]
• ${\displaystyle A+\operatorname {core} B=\operatorname {core} (A+B)}$  if ${\displaystyle B=\operatorname {core} B.}$ ^[7]

Both the core and the algebraic closure of a convex set are again convex.^[5] If ${\displaystyle C}$  is convex, ${\displaystyle c\in \operatorname {core} C,}$  and ${\displaystyle b\in \operatorname {acl} _{X}C}$  then the line segment ${\displaystyle [c,b):=c+[0,1)b}$  is contained in ${\displaystyle \operatorname {core} C.}$ ^[5]

Relation to topological interior edit

Let ${\displaystyle X}$  be a topological vector space, ${\displaystyle \operatorname {int} }$  denote the interior operator, and ${\displaystyle A\subseteq X}$  then:

• ${\displaystyle \operatorname {int} A\subseteq \operatorname {core} A}$ 
• If ${\displaystyle A}$  is nonempty convex and ${\displaystyle X}$  is finite-dimensional, then ${\displaystyle \operatorname {int} A=\operatorname {core} A.}$ ^[1]
• If ${\displaystyle A}$  is convex with non-empty interior, then ${\displaystyle \operatorname {int} A=\operatorname {core} A.}$ ^[8]
• If ${\displaystyle A}$  is a closed convex set and ${\displaystyle X}$  is a complete metric space, then ${\displaystyle \operatorname {int} A=\operatorname {core} A.}$ ^[9]

If ${\displaystyle M=\operatorname {aff} (A-A)}$  then the set ${\displaystyle \operatorname {aint} _{M}A}$  is denoted by ${\displaystyle {}^{i}A:=\operatorname {aint} _{\operatorname {aff} (A-A)}A}$  and it is called the relative algebraic interior of ${\displaystyle A.}$ ^[7] This name stems from the fact that ${\displaystyle a\in A^{i}}$  if and only if ${\displaystyle \operatorname {aff} A=X}$  and ${\displaystyle a\in {}^{i}A}$  (where ${\displaystyle \operatorname {aff} A=X}$  if and only if ${\displaystyle \operatorname {aff} (A-A)=X}$ ).

If ${\displaystyle A}$  is a subset of a topological vector space ${\displaystyle X}$  then the relative interior of ${\displaystyle A}$  is the set

If ${\displaystyle A}$  is a subset of a topological vector space ${\displaystyle X}$  then the quasi relative interior of ${\displaystyle A}$  is the set

In a Hausdorff finite dimensional topological vector space, ${\displaystyle \operatorname {qri} A={}^{i}A={}^{ic}A={}^{ib}A.}$ 

• Bounding point – Mathematical concept related to subsets of vector spaces
• Interior (topology) – Largest open subset of some given set
• Order unit – Element of an ordered vector space
• Quasi-relative interior – Generalization of algebraic interior
• Radial set
• Relative interior – Generalization of topological interior
• Ursescu theorem – Generalization of closed graph, open mapping, and uniform boundedness theorem

Bibliography edit

• Aliprantis, Charalambos D.; Border, Kim C. (2006). Infinite Dimensional Analysis: A Hitchhiker's Guide (Third ed.). Berlin: Springer Science & Business Media. ISBN 978-3-540-29587-7. OCLC 262692874.
• Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.
• Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.
• Schechter, Eric (1996). Handbook of Analysis and Its Foundations. San Diego, CA: Academic Press. ISBN 978-0-12-622760-4. OCLC 175294365.
• Zălinescu, Constantin (30 July 2002). Convex Analysis in General Vector Spaces. River Edge, N.J. London: World Scientific Publishing. ISBN 978-981-4488-15-0. MR 1921556. OCLC 285163112 – via Internet Archive.