Convexity

A term used in various branches of mathematics and indicating properties which generalize some properties of convex sets (cf. Convex set) in Euclidean spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026400/c0264001.png" />. The applicability of a number of methods of study is connected with the term "convexity" .

The following two basic definitions are almost equivalent in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026400/c0264002.png" />. A set is convex: 1) if it is the intersection of open half-spaces; or 2) if it contains with two arbitrary points the segment connecting these points. Both definitions of convexity are applied to the case of vector spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026400/c0264004.png" />.

Definition 2) is generalized to include sets in spaces with geodesics (spaces with a connection; locally compact metric spaces, in particular Riemannian and Finsler spaces). The geodesics play the role of segments, but if the two points can be connected by non-unique geodesics or by shortest curves, the concept of convexity branches into several possible meanings. In Riemannian geometry, in particular, the following modifications of convexity are employed [1], [2]:

a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026400/c0264005.png" /> is convex if any two points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026400/c0264006.png" /> can be connected by a unique shortest curve and if this shortest curve is contained in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026400/c0264007.png" />;

a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026400/c0264008.png" /> is locally convex if each point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026400/c0264009.png" /> has a neighbourhood in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026400/c02640010.png" /> which is convex in the sense of ;

a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026400/c02640011.png" /> is weakly convex if any two points can be connected by at least one shortest curve lying in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026400/c02640012.png" />; and

a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026400/c02640013.png" /> is absolutely convex if all geodesics connecting any two points are located in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026400/c02640014.png" />.

The boundary (or part of the boundary) of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026400/c02640015.png" />-dimensional convex body in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026400/c02640016.png" /> is called a convex hypersurface; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026400/c02640017.png" />, it is called a convex surface, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026400/c02640018.png" />, it is said to be a convex curve.

For a function of a real variable convexity signifies the convexity of its supergraph (cf. Convex function (of a real variable)). The convexity of a functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026400/c02640019.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026400/c02640020.png" /> is defined in a similar manner (cf. Convex functional).

For convex sets in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026400/c02640021.png" /> one can speak of the convexity of a family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026400/c02640022.png" /> of sets: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026400/c02640023.png" /> is convex if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026400/c02640024.png" /> provided that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026400/c02640025.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026400/c02640026.png" />. Convex (and concave) functionals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026400/c02640027.png" /> can be defined on convex families <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026400/c02640028.png" />. Convexity of a functional is defined by the requirement

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026400/c02640029.png" />

The term "convexity" as applied to univalent functions of a complex variable has a special meaning — the property of mapping the unit disc onto a convex domain (cf. Convex function (of a complex variable)).

The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026400/c02640031.png" />-convexity of a compactum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026400/c02640032.png" />, meaning that each point with distance to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026400/c02640033.png" /> less than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026400/c02640034.png" /> has a unique nearest point in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026400/c02640035.png" />, is one generalization of convexity in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026400/c02640036.png" /> which has been studied [4], [5].

In the theory of linear differential operators the term "convexity" is connected with certain properties of homology groups [6]. This is connected with the possibility of touching the boundary from inside a domain by a hypersurface for which a certain number of principal curvatures is positive. In the theory of functions of several complex variables an important role is played by holomorphic convexity, which is connected with the impossibility of touching the boundary of a domain from inside with an analytic surface [7]. The last-named concept is a special case of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026400/c02640038.png" />-convexity [7]. To the concept of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026400/c02640039.png" />-convexity one can transfer many given convexity properties.

The concept of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026400/c02640041.png" />-convexity is employed in convex analysis; it is a generalization of the representability of a convex function as the supremum of a family of linear functions [8].

In the theory of metric spaces, (Menger) convexity of a metric is defined as follows: For any points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026400/c02640042.png" /> there exists a third point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026400/c02640043.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026400/c02640044.png" /> [9]. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026400/c02640046.png" />-convexity of a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026400/c02640047.png" /> means that moreover <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026400/c02640048.png" /> belongs to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026400/c02640049.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026400/c02640050.png" />. Definitions of convexity in ordered spaces (cf. Convex subgroup) are very similar.

To almost each definition of convexity there corresponds a concept of local convexity. However, for the class of locally convex topological vector spaces, the term "local convexity" has a special meaning, namely, that for each point there exists a basis system of convex neighbourhoods.

References

[1]	D. Gromoll, W. Klingenberg, W. Meyer, "Riemannsche Geometrie im Grossen" , Springer (1968)
[2]	A.D. Aleksandrov, V.A. Zalgaller, "Intrinsic geometry of surfaces" , Transl. Math. Monogr. , Amer. Math. Soc. (1967) (Translated from Russian)
[3]	H. Hadwiger, "Vorlesungen über Inhalt, Oberfläche und Isoperimetrie" , Springer (1957)
[4]	H. Federer, "Curvature measures" Trans. Amer. Math. Soc. , 93 (1959) pp. 418–491
[5]	Yu.G. Reshetnyak, "On a generalization of convex surfaces" Mat. Sb. , 40 (82) : 3 (1956) pp. 381–398 (In Russian)
[6]	V.P. Palamodov, "Linear differential operators with constant coefficients" , Springer (1970) (Translated from Russian)
[7]	V.S. Vladimirov, "Methods of the theory of functions of several complex variables" , M.I.T. (1966) (Translated from Russian)
[8]	S.S. Kutateladze, A.M. Rubinov, "Minkowski duality and applications" Russian Math. Surveys , 27 : 3 (1972) pp. 137–191 Uspekhi Mat. Nauk , 27 : 3 (1972) pp. 127–176
[9]	L. Danzer, B. Grünbaum, V.L. Klee, "Helly's theorem and its relatives" V.L. Klee (ed.) , Convexity , Proc. Symp. Pure Math. , 7 , Amer. Math. Soc. (1963) pp. 101–180

Comments

Definitions 1) and 2) are equivalent for most, but not all, sets. E.g., a convex set in the sense of 2), maximal with the property of not containing a given point, does not satisfy 1). An explicit example is the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026400/c02640051.png" />, the unit square minus one half of an edge. The space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026400/c02640052.png" /> satisfies 2), but does not satisfy 1).

A more usual definition of convexity of a metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026400/c02640053.png" />: For any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026400/c02640054.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026400/c02640055.png" />, and any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026400/c02640056.png" /> there is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026400/c02640057.png" /> so that

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026400/c02640058.png" />

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026400/c02640059.png" />

General results on convexity can be found in [a1], [a6]. Convexity also plays an important role in optimization theory, variational problems and the geometry of numbers (cf. [a2], [a3], [a5], [a7] and [a8]).

For holomorphic convexity cf. Stein manifold; the phrase "K-convex" is also used to denote the property of a set that each connected component is convex ([a4]); cf. Geometry of immersed manifolds for the notion of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026400/c02640061.png" />-convex surface.

References

[a1]	E.M. Alfsen, "Compact convex sets and boundary integrals" , Springer (1971)
[a2]	J.T. Marti, "Konvexe Analysis" , Birkhäuser (1977)
[a3]	R.T. Rockafellar, "Convex analysis" , Princeton Univ. Press (1969)
[a4]	F.A. Valentine, "Convex sets" , McGraw-Hill (1964)
[a5]	I. Ekeland, R. Teman, "Convex analysis and variational problems" , North-Holland (1976)
[a6]	G. Choquet, "Lectures on analysis" , 1–3 , Benjamin (1969) (Translated from French)
[a7]	P.M. Gruber, C.G. Lekkerkerker, "Geometry of numbers" , North-Holland (1987) (Updated reprint)
[a8]	V. Barbu, Th. Precupanu, "Convexity and optimization in Banach spaces" , Reidel (1986)