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'''Complex convexity''' is a general term in [[complex geometry]]. | title = Complex convexity and analytic functionals ...

* general convexity ...title=52Axx Mathematics Subject Classification MSC2010, entry 52A "General convexity"]</ref> ...

=== Relation to other types of "convexity" === ...s this function for the case where <math>k=0.2</math>. This example may be generalized to two variables as: ...

...th> (for <math display="inline">p > 1</math>) and so, by the definition of convexity, ===Proof by a direct convexity argument=== ...

{{for|the unrelated generalization of convexity used in the calculus of variations|Quasiconvexity (calculus of variations)} Quasiconvexity is a more general property than convexity in that all [[convex function]]s are also quasiconvex, but not all quasicon ...

The exponent <math>r</math> can be generalized to an arbitrary real number as follows: if <math>x>-1</math>, then This generalization can be proved by convexity (see below) or by comparing [[derivative]]s. The strict versions of these i ...

; [[Generalized mean]]: A generalization of the [[Pythagorean means]], specified by an expo ; [[Quasi-arithmetic mean]]: A generalization of the [[generalized mean]], specified by a [[continuous function|continuous]] [[injective funct ...

.../ref><ref>{{cite journal|last1=Kjeldsen|first1=Tinne Hoff|title=History of Convexity and Mathematical Programming|journal=Proceedings of the International Congr ...the [[interval (mathematics)|interval]] {{math|[0, 1]}}. This implies that convexity is invariant under [[affine transformation]]s. Further, it implies that a c ...

...<math>f( r x + (1 - r) y ) \leq r f(x) + (1 - r) f(y)</math>|{{EquationRef|Convexity ≤}}|LnSty=1px dashed black}} ...emains true of <math>f</math> when the defining inequality ({{EquationNote|Convexity ≤}}) is replaced by the strict inequality ...

...[Bohr–Mollerup theorem]] for the gamma function where strictly logarithmic convexity on <math>\mathbb{R}^{+}</math> is demanded additionally. The case {{math|'' is completely monotone. The convexity inequality {{math|''e^t'' ≥ 1 + ''t''}} implies that ...

...t manifold]]. This allows many techniques of [[optimization theory]] to be generalized to Bregman divergences, geometrically as generalizations of [[least squares ...math> for all <math>p</math>, <math>q</math>. This is a consequence of the convexity of <math>F</math>. ...

== Schur convexity == Majorization can be generalized to the [[Lorenz curve|Lorenz ordering]], a partial order on [[cumulative di ...

...second statement can also be modified to get the definition of {{em|strict convexity}}, where the latter is obtained by replacing <math>\,\leq\,</math> with the ...for every fixed <math>x_2</math> (or vice versa). This characterization of convexity is quite useful to prove the following results. ...

The subadditivity axiom may then be replaced by the following '''strong convexity condition''': ...s the '''fundamental tensor''' of {{math|''F''}} at {{math|''v''}}. Strong convexity of {{math|''F''}} implies the subadditivity with a strict inequality if {{m ...

...eneralized version of the theorem is sufficient to prove [[Convex function|convexity]] when the one-sided derivatives are [[monotonically increasing]]:<ref>{{ci == Proof of the generalized version == ...

[[File:Convex 01.ogg|thumb|right|Visualizing convexity and Jensen's inequality]] ...) probability. In the probabilistic setting, the inequality can be further generalized to its ''full strength''. ...

| known_for = [[Kostant's convexity theorem]]<br>[[Kostant partition function]]<br>[[Kostant polynomial]]<br>[[ *{{cite journal|title=Lie algebra cohomology and the generalized Borel-Weil theorem|journal=Annals of Mathematics|year=1961|volume=74|issue= ...

...f the derivative of f(x) is also of interest, as it is used in showing the convexity of the [[Legendre transformation|Legendre transform]]. These formulas are generalized by the [[Faà di Bruno's formula]]. ...

...ories of [[abstract polytope]]s and [[star polytope]]s, the requirement of convexity is relaxed. One precise combinatorial concept that generalizes some earlier The notion of a face can be generalized from convex polytopes to all [[convex set]]s, as follows. Let <math>C</math ...

...name=":0">{{Cite journal |last=Grantier |first=Bruce J. |date=1988 |title=Convexity and Bond Performance: The Benter the Better |url=https://www.tandfonline.co ...journal |last=Handforth |first=Frank |date=2004-01-01 |title=Duration and Convexity of Mortgages in the Context of Real Estate Investment Analysis |url=https:/ ...