  The space $\mathcal{H}$ of "almost calibrated" $(1,1)$ forms on a compact
K\"ahler manifold plays an important role in the study of the deformed
Hermitian-Yang-Mills equation of mirror symmetry as emphasized by recent work
of the second author and Yau, and is related by mirror symmetry to the space of
positive Lagrangians studied by Solomon. This paper initiates the study of the
geometry of $\mathcal{H}$. We show that $\mathcal{H}$ is an infinite
dimensional Riemannian manifold with non-positive sectional curvature. In the
hypercritical phase case we show that $\mathcal{H}$ has a well-defined metric
structure, and that its completion is a ${\rm CAT}(0)$ geodesic metric space,
and hence has an intrinsically defined ideal boundary. Finally, we show that in
the hypercritical phase case $\mathcal{H}$ admits $C^{1,1}$ geodesics,
improving a result of the second author and Yau. Using results of
Darvas-Lempert we show that this result is sharp.
