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We construct a hereditarily indecomposable Banach space with dual space isomorphic to$ℓ$_{1}. Every bounded linear operator on this space is expressible as λ$I$+$K$, with λ a scalar and$K$compact.

A simplicial complex Δ is called$flag$if all minimal nonfaces of Δ have at most two elements. The following are proved: First, if Δ is a flag simplicial pseudomanifold of dimension$d$−1, then the graph of Δ (i) is (2$d$−2)-vertex-connected and (ii) has a subgraph which is a subdivision of the graph of the$d$-dimensional cross-polytope. Second, the$h$-vector of a flag simplicial homology sphere Δ of dimension$d$−1 is minimized when Δ is the boundary complex of the$d$-dimensional cross-polytope.

We establish local higher integrability estimates for upper gradients of vector-valued parabolic quasi-minimizers in metric measure spaces, satisfying a doubling property and supporting a weak Poincaré inequality.