Let X be a real vector space, and Φ be a subspacePlanetmathPlanetmathPlanetmath of linear functionalsMathworldPlanetmathPlanetmath on X.

For any set SX, its anti-cone S+, with respect to Φ, is the set

S+={ϕΦ:ϕ(x)0, for all xS}.

The anti-cone is also called the dual cone.


The anti-cone operationMathworldPlanetmath is generally applied to subsets of X that are themselves cones. Recall that a cone in a real vector space generalize the notion of linear inequalities in a finite number of real variables. The dual cone provides a natural way to transfer such inequalitiesMathworldPlanetmath in the original vector spaceMathworldPlanetmath to its dual spaceMathworldPlanetmathPlanetmathPlanetmath. The concept is useful in the theory of duality.

The set Φ in the definition may be taken to be any subspace of the algebraic dual space X*. The set Φ often needs to be restricted to a subspace smaller than X*, or even the continuous dual space X, in order to obtain the nice closure and reflexivityMathworldPlanetmath properties below.

Basic properties

Property 1.

The anti-cone is a convex cone in Φ.


If ϕ(x) is non-negative, then so is tϕ(x) for t>0. And if ϕ1(x),ϕ2(x)0, then clearly (1-t)ϕ1(x)+tϕ2(x)0 for 0t1. ∎

Property 2.

If KX is a cone, then its anti-cone K+ may be equivalently characterized as:

K+={ϕΦ:ϕ(x) over xK is bounded below}.

It suffices to show that if infxKϕ(x) is bounded below, then it is non-negative. If it were negative, take some xK such that ϕ(x)<0. For any t>0, the vector tx is in the cone K, and the function value ϕ(tx)=tϕ(x) would be arbitrarily large negative, and hence unboundedPlanetmathPlanetmath below. ∎

Topological properties

AssumptionsPlanetmathPlanetmath. Assume that Φ separates points of X. Let X have the weak topology generated by Φ, and let Φ have the weak-* topology generated by X; this makes X and Φ into Hausdorff topological vector spacesMathworldPlanetmath.

Vectors xX will be identified with their images x^ under the natural embedding of X in its double dual space.

The pairing (X,Φ) is sometimes called a dual pair; and (Φ,X), where X is identified with its image in the double dual, is also a dual pair.

Property 3.

S+ is weak-* closed.


Let {ϕα}Φ be a net converging to ϕ in the weak-* topology. By definition, x^(ϕα)=ϕα(x)0. As the functionalMathworldPlanetmathPlanetmath x^ is continuousMathworldPlanetmath in the weak-* topology, we have x^(ϕα)x^(ϕ)0. Hence ϕS+. ∎

Property 4.



The inclusion S¯+S+ is obvious. And if ϕ(x)0 for all xS, then by continuity, this holds true for xS¯ too — so S¯+S+. ∎

Properties involving cone inclusion

Property 5 (Farkas’ lemma).

Let KX be a weakly-closed convex cone. Then xK if and only if ϕ(x)0 for all ϕK+.


That ϕ(x)0 for ϕK+ and xK is just the definition. For the converseMathworldPlanetmath, we show that if xXK, then there exists ϕK+ such that ϕ(x)<0.

If K=, then the desired ϕΦ=K+ exists because Φ can separate the points x and 0. If K, by the hyperplane separation theorem, there is a ϕΦ such that ϕ(x)<infyKϕ(y). This ϕ will automatically be in K+ by Property 2. The zero vector is the weak limit of ty, as t0, for any vector y. Thus 0K, and we conclude with infyKϕ(y)0. ∎

Property 6.

K++=K¯ for any convex cone K. (The anti-cone operation on K+ is to be taken with respect to X.)


We work with K¯, which is a weakly-closed convex cone. By Property 5, xK¯ if and only if ϕ(x)0 for all ϕK¯+=K+. But by definition of the second anti-cone, x^(K+)+ if and only if ϕ(x)=x^(ϕ)0 for all ϕK+. ∎

Property 7.

Let K and L be convex cones in X, with K weakly closed. Then K+L+ if and only if KL.



  • 1 B. D. Craven and J. J. Kohila. “GeneralizationsPlanetmathPlanetmath of Farkas’ TheoremMathworldPlanetmath.” SIAM Journal on Mathematical Analysis. Vol. 8, No. 6, November 1977.
  • 2 David G. Luenberger. Optimization by Vector Space Methods. John Wiley & Sons, 1969.
Title anti-cone
Canonical name Anticone
Date of creation 2013-03-22 17:20:48
Last modified on 2013-03-22 17:20:48
Owner stevecheng (10074)
Last modified by stevecheng (10074)
Numerical id 8
Author stevecheng (10074)
Entry type Definition
Classification msc 46A03
Classification msc 46A20
Synonym anticone
Synonym dual cone
Related topic GeneralizedFarkasLemma