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For closed 3-manifolds, Heegaard Floer homology is related to the Thurstonnorm through results due to Ozsv\'ath and Szab\'o, Ni, and Hedden. For example,given a closed 3-manifold Y, there is a bijection between vertices of theHF^+(Y) polytope carrying the group Z and the faces of the Thurston norm unitball that correspond to fibrations of Y over the unit circle. Moreover, theThurston norm unit ball of Y is dual to the polytope of \underline{\hfhat}(Y).  We prove a similar bijection and duality result for a class of 3-manifoldswith boundary called sutured manifolds. A sutured manifold is essentially acobordism between two surfaces R_+ and R_- that have nonempty boundary. We showthat there is a bijection between vertices of the sutured Floer polytopecarrying the group Z and equivalence classes of taut depth one foliations thatform the foliation cones of Cantwell and Conlon. Moreover, we show that afunction defined by Juh\'asz, which we call the geometric sutured function,isanalogous to the Thurston norm in this context. In some cases, this function isan asymmetric norm and our duality result is that appropriate faces of thisnorm's unit ball subtend the foliation cones.  An important step in our work is the following fact: a sutured manifoldadmits a fibration or a taut depth one foliation whose sole compact leaves areexactly the connected components of R_+ and R_-, if and only if, there is asurface decomposition of the sutured manifold resulting in a connected productmanifold.

The sutured Floer polytope and taut depth one foliations