Separation, convexity and polarity in the space of normlinear functions

The paper studies separation properties for subsets of the space (Formula presented.) 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 (Formula presented.). 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 (Formula presented.), seen as a function defined on (Formula presented.). Since in this case the evaluation functionals are linear, separation describes a special subclass of convex sets in (Formula presented.). We characterize X-convex sets by exploiting the isomorphism between (Formula presented.) and the space (Formula presented.). We also study polarity relations defined on (Formula presented.). Polar sets are X-convex. And we describe what further properties are needed in order to make a subset (Formula presented.) the polar, or the reverse polar, of some set in X. For polar sets (Formula presented.) it is possible to deduce the unique closed, radiant prepolar set (Formula presented.). To conclude, we emphasize some connections between upward (downward) X-convex sets in Y and sets of upper (resp. lower) bounds in Y.