Complex geometry is a cornerstone of contemporary mathematics; convex geometry, by contrast, has remained fairly niche since its modern inception in the early 20th century. Nonetheless, the two subjects are related by numerous extremely deep correspondences. Even more bizarrely, these correspondences tend to end up being more "useful" on the complex side, suggesting that there is some "structure" on the complex side that is somehow more naturally analyzed through its convex counterpart. In this talk, we will sample some results from this fascinating history before discussing a "convex" approach to studying Kähler–Einstein metrics.