Separation, convexity and polarity in the space of normlinear functions

Abstract

The paper studies separation properties for subsets of the space $\mathcal{N}$ of normlinear functions on the Banach space X, i.e. the sum of a linear function and a multiple of the norm. We use as separating functionals the ones that are generated by the duality pairing $[x,p]=p\left(x\right)=〈x,a〉+\delta \Vert x\Vert$. We show that normlinear functions can be used to separate points from radiant or coradiant sets in X. Then we pass to separation defined by means of the pairing $[x,\cdot ]$, seen as a function defined on $\mathcal{N}$. Since in this case the evaluation functionals are linear, separation describes a special subclass of convex sets in $\mathcal{N}$. We characterize X-convex sets by exploiting the isomorphism between $\mathcal{N}$ and the space $Y=X^{\ast }\times \mathbb{R}$. We also study polarity relations defined on $\mathcal{N}$. Polar sets are X-convex. And we describe what further properties are needed in order to make a subset $C\subset Y$ the polar, or the reverse polar, of some set in X. For polar sets $C\subset Y,$ it is possible to deduce the unique closed, radiant prepolar set $U\subset X$. To conclude, we emphasize some connections between upward (downward) X-convex sets in Y and sets of upper (resp. lower) bounds in Y.

Keywords:

• Nonlinear duality
• separation
• normlinear functions
• polarity
• radiant sets

AMS CLASSIFICATIONS:

• 52A30
• 52A07

Acknowledgments

I wish to thank an anonymous referee, whose careful reading and clever criticisms allowed me to correct some inaccuracies and improve the presentation of the results.

Disclosure statement

No potential conflict of interest was reported by the author(s).