One cake, two people who want different parts of it — she loves the frosting end, he loves the fruit. Split it down the middle and both feel cheated. Drag the cut until each of them, by their OWN valuation, believes they got at least half. That line exists, and it is rarely the middle.
Each party has a valuation density over the cake (Ann's peaks at the frosting, Ben's at the fruit), normalised to total value 1. For a cut at x, Ann's share is the integral of her density on [0,x] and Ben's on [x,1] — computed live. An envy-free cut is one where each party values their own piece at ≥ ½ by their OWN valuation; because they value regions differently, such a cut exists and is generally NOT the midpoint (the instrument finds it). Real fair-division / cake-cutting theory (Steinhaus).
Two parties and a one-dimensional cake with Gaussian tastes; real fair division handles many parties, indivisible goods, and strategic mis-reporting (a whole field). The core — a division fair by each party's own measure exists and is rarely the equal split — is the exact, honest content.