Some density properties of the closed unit ball of \(L_1\) (Q837639)

Let \((E,\tau)\) be a topological space. A curve is a continuous mapping \(\gamma:[0,1]\to E\). Now assume that \((E,\tau)\) is a topological vector space, let \(A\subset E\) and let \(V\) be a neighborhood of \(0\). The curve \(\gamma\) is said to be \(V\)-dense in \(A\) if \(\gamma([0,1])\subset A\) and, for any \(x\in A\), some point \(\gamma(t)\) fulfills \(x-\gamma(t)\in V\). If \(A\) contains a \(V\)-dense curve for all \(V\), \(A\) is called densifiable. It is observed that all bounded convex subsets of locally convex topological vector spaces are densifiable when endowed with the weak topology. In particular, the unit ball of any normed space is weakly densifiable. The author then turns to the special case of the unit ball of \(L_1[0,1]\) and shows that the degree of weak densifiability is 0. The non-metric character of this situation is then described: Let \(\gamma\) be some curve in \((B_{L_1},\text{weak})\) and \(0<\lambda<1\). Then there exists a weak \(0\)-neighborhood of spherical order \(\lambda\) such that \(\gamma\) is not \(V\)-dense in \(B_{L_1}\). In the last section of the paper, an approach towards the solution of a global optimization problem on a set \(D\) is performed through solutions of one-dimensional global optimization problems on each curve that densifies \(D\).