Zur theorie der polarfunktionale

On Monge–Ampère type equations arising in optimal transportation problems

“…However, it can be readily checked that the same argument works for general g. To this end, we only need to note that from the assumptions on g we have spt(ω c (g, u)) = spt(ω c (u)) and if E, F ⊂ R n are two measurable sets with E ⊂ F and E g < ∞, then E g = F g if and only if |E| = |F|. By adding a constant to v if necessary, we can assume without loss of generality that 8 We havex ∈ D δ , and w δ (…”

Duality for Nonconvex Approximation and Optimization

“…Later, the duality results obtained in this direction were extended to duality results for nonconvex problems, based on generalizations of convexity and of the methods of convex analysis by Elster and Nehse [60], Balder [13], Lindberg [134], Dolecki and Kurcyusz [48], Dolecki [47], and others.…”

“…The equivalence 1° ^ 3° of Corollary 1.2, Lemma 1.6, and Remark 1.4 have been given in [210, The pseudoconjugates (1.220) and the semiconjugates (1.221) have been introduced and studied in [224] and, respectively, [233]; actually, the semiconjugate has been introduced in order to obtain a conjugate whose biconjugate is the lower semicontinuous quasi-convex hull (formula (1.222)). For some other conjugates of functions f:X->R, useful for duality in convex and quasi-convex optimization, and for some generaUzations, see [234], [243], [254], [260], Atteia [5], Atteia and El Qortobi [6], Crouzeix [34], [35], Elster and Gopfert [59], Elster and Nehse [60], Elster and Wolf [62]- [64], Martinez-Legaz [138]- [141], Passy and Prisman [171], [172], Penot [176], Penot and Voile [177], Voile [287], [288].…”

Mass Transportation Problems

“…For the literature on c-convexity and c-conjugation we refer to Dietrich (1988) and Elster and Nehse (1974).…”