Geodesic
A new relationship between decomposability and convexity
Commentationes Mathematicae Universitatis Carolinae, Tome 47 (2006) no. 3, pp. 457-466.

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In the present work we prove that, in the space of Pettis integrable functions, any subset that is decomposable and closed with respect to the topology induced by the so-called Alexiewicz norm $\left| \left\|\cdot \right\| \right|$ \big(where $\left| \left\| f\right\| \right| =\sup_{[a,b] \subset [0,1]} \big\| \int_{a}^{b}f(s) ds \big\|$\big) is convex. As a consequence, any such family of Pettis integrable functions is also weakly closed.
Classification : 28B05, 46A20, 46A55, 46E30, 46E40, 46G10, 52A07, 54A10
Keywords: Pettis integral; decomposable set; convex set; Alexiewicz norm 
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     title = {A new relationship between decomposability and convexity},
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Satco, Bianca. A new relationship between decomposability and convexity. Commentationes Mathematicae Universitatis Carolinae, Tome 47 (2006) no. 3, pp. 457-466. http://geodesic.mathdoc.fr/item/CMUC_2006__47_3_a7/