Intersection properties for cones of monotone and convex functions in scale of Lipschitz spaces (Q2720289)

The cone intersection property and its modifications for cones of univariate monotone and convex functions with respect to the scale of Lipschitz (Nikol'skii-Besov) spaces is studied. Let \(M\) be the cone \(\{f:[0,1)\to {\mathbb{R}}\), \(f\geqslant 0\), \(f\uparrow\}\) and \(C\) be the cone \(\{f:[0,1)\to {\mathbb{R}}\), \(f\geqslant 0\), \(f\uparrow\), \(f\)~convex\(\}\). Then:NEWLINENEWLINENEWLINE(i) If \(0<\alpha <1/p\) and \(1\leqslant p<\infty\) then \(M\) has the intersection property (IP) with respect to \((L_p,H^\alpha_p)\);NEWLINENEWLINENEWLINE(ii)~If \(\alpha \geqslant 1/p\), \(1\leqslant p<\infty\) or \(\alpha >0\), \(p=\infty\), then \(M\) does not have neither the weak intersection property (WIP), nor the IP with respect to \((L_p,\dot H^\alpha_p)\);NEWLINENEWLINENEWLINE(iii) \(C\) has the IP with respect to \((L_p,H^\alpha_p)\), \(p\in [1,\infty]\), if \(0<\alpha <1\);NEWLINENEWLINENEWLINE(iv)~\(C\) does not have the WIP with respect to \((L_p,\dot H^\alpha_p)\), if \(\alpha >1\).