立命館大学数理科学科

日時： 5月27日(金) 
場所： 立命館大学 BKC フォレストハウス1階 F105
時間： 16:30 — 18:00
講演者： 尾張 圭太 氏（立命館大学）, 多羅間 大輔 氏（立命館大学）
1. 16:30-17:10
講演者： 尾張 圭太 氏（立命館大学）

講演題目： On Convex Functions on Orlicz Spaces with $\Delta_2$-Conjugates

アブストラクト： We show that in an Orlicz space \(L^\Phi\) with the conjugate Young function \(\Psi\) being \(\Delta_2\) (so \(L^\Phi\) is the dual of \(L^\Psi\)), a proper convex function has a \(\langle L^\Phi,L^\Psi\rangle\)-dual representation iff it is order lower semicontinuous; more precisely, a convex set \(C\subset L^\Phi\) is \(\sigma(L^\Phi,L^\Psi)\)-closed iff for each order interval \([-\eta,\eta]=\{\xi:-\eta\leq \xi\leq \eta\}\) (\(0\leq \eta\in L^\Phi\)), the intersection \(C\cap [-\eta,\eta]\) is closed in \(L^0\). The result is based on the following technical lemma: for any norm bounded sequence \((\xi_n)_n\) in \(L^\Phi\) which converges in probability to \(0\), there exist forward convex combinations \(\zeta_n\in \mathrm{conv}(\xi_n,\xi_{n+1},…)\) as well as an element \(\eta\in L^\Phi_+\) such that \(\zeta_n\rightarrow 0\) and \(|\zeta_n|\leq \eta\). We show also that a finite-valued convex function on \(L^\Phi\) is \(\tau(L^\Phi,L^\Psi)\)-continuous iff it is sequentially \(\tau(L^\Phi,L^\Psi)\)-continuous on order intervals, and the condition is equivalent to the order continuity of the function.