Abstract: Convex geometry has recently appeared in several related guises in homological algebra: as g-fans in the silting theory of finite-dimensional algebras, as wall-and-chamber structures for abelian length categories, and as scattering diagrams in Bridgeland stability theory. I will discuss joint work with Nathan Broomhead, David Pauksztello and David Ploog on a general construction which we hope provides a natural and unifying context. The input data for our construction is a triangulated category D and a finite rank lattice quotient L of its Grothendieck group. Let V =Hom(L,R) be the dual real vector space. Each heart H in D determines a closed convex `heart cone' in V, and the heart cones of H and all its forward tilts form a `heart fan' in V. The simplest case is when the heart H is `algebraic', i.e. is a length category with finitely many simple objects, in which case the heart cone is simplicial and the heart fan is complete. The heart fans for all hearts in D can be assembled into a `multifan' whose tangent space can be interpreted as the space of `lax stability functions' on D with values in the complexification of V. The complex manifold Stab(D) of Bridgeland stability conditions embeds as an open subset of this tangent space. There is a nice interplay between the homological algebra of D and the convex geometry of the multifan, and I will try to use the latter to illustrate the former as much as possible, keeping the algebraic prerequisites to a minimum.