Sufficient conditions for existence of extrema in linear programming

AbstractIt is shown that, on a closed convex subset X of a real Hausdorff locally convex space E, a continuous linear functional x′ on E has an extremum at an extreme point of X, provided X contains no line and X ∩ (x′)−1 (λ0) is non-empty and weakly compact for some real λ0. It is also shown that any weakly locally compact closed convex subset of E that contains no line is the sum of its asymptotic cone and the closed convex hull of its extreme points