Locally Uniformly Quasi-Convex Programming
Carl G. Looney · SIAM Journal on Applied Mathematics · 1975
If X is a convex subset of a locally convex Hausdorif topological vector space $( {E,\tau } )$ and f is a real-valued $\tau $-l.s.c. quasi-convex functional on X, then f is also weakly l.s.c. on X and thus attains its infimum on X whenever X is weakly compact. Further, if f is locally uniformly quasi-convex and E is a Banach space, then any minimizing sequence $( {x_n } )$ in X converges in norm to a unique point $x'$ whenever f attains its infimum (with X not necessarily weakly compact), and $\inf f( X ) = f( {x'} )$. An important application is any locally uniformly convex norm. Moreover, it is known that any weakly compact convex set X in a Banach space determines a locally uniformly convex norm on the subspace generated by X.