Therefore Λ := — Λ is a positive linear form fulfilling Λ(X — Y) ≤ ρ+(X, Y — X) ≤ ρ(Y) — ρ(X) for Y ∈ X,
and Λ(Y) ≤ π(Y) for Y ∈ C. This implies Λ∣C = π due to linearity of Λ∣C and π. Furthermore we have shown
βρ(Λ) = —Λ(X ) — ρ(X ).
For the proof of the equivalence stated in Proposition 1.1 note that the only if part is obvious, the if part follows
immediately from the Fenchel-Moreau theorem (cf. [2], Theorem 4.2.2), observing that βρ (Λ) < ∞ only if Λ is
positive linear and extends π (see also Proposition 3.9 in [9]). This completes the proof. H
Acknowlegdements:
The author would like to thank Freddy Delbaen and Alexander Schied for helpful hints. He is also indepted to
Heinz Konig for good advice and continuous exchange of ideas from the fields of measure theory and superconvex
analysis.
References
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