# Additive selection

A mapping $ s : H \rightarrow G $
associated with a set-valued function $ F $
from an (Abelian) semi-group $ H $
to subsets of an (Abelian) semi-group $ G $
which is a homomorphism (of semi-groups) and a selection of $ F $.
If $ G \subset 2 ^ {H} $
and $ F $
is the identity transformation on $ G $,
then $ s $
is said to be an additive selection on $ G $.
An archetypical example of an additive selection is the mapping which subordinates to each non-empty compact set in $ \mathbf R $
its maximal element. In $ \mathbf R ^ {n} $,
similar selections can be defined by means of lexicographic orders, see [a1] and Lexicographic order). They are Borel measurable but not continuous with respect to the Hausdorff metric (for $ n > 1 $).
A Lipschitz-continuous additive selection on the family of convex bodies in $ \mathbf R ^ {n} $
is given by associating with each convex body its Steiner point, see [a2]. No such selections can exist in infinite dimensions, see [a3], [a4].

#### References

[a1] | R. Živaljević, "Extremal Minkowski additive selections of compact convex sets" Proc. Amer. Math. Soc. , 105 (1989) pp. 697–700 |

[a2] | R. Schneider, "Convex bodies: the Brunn–Minkowski theory" , Cambridge Univ. Press (1993) |

[a3] | R.A. Vitale, "The Steiner point in infinite dimensions" Israel J. Math. , 52 (1985) pp. 245–250 |

[a4] | K. Przesławski, D. Yost, "Continuity properties of selectors and Michael's theorem" Michigan Math. J. , 36 (1989) pp. 113–134 |

**How to Cite This Entry:**

Additive selection.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Additive_selection&oldid=45029