Picture a triangular dartboard where each corner is a stock. A point inside shows how the money is split: for example, shares of ⅓, ⅓, ⅓ mark the exact center. Heading for the center feels safe — after all, you’re moving away from the edges where you could lose everything. But the shape of this board is deceptive: sometimes the center is an invisible pit.

The trouble lies in the rule for combining shares. If this rule is curved (bowing upward in the middle), then averaging two portfolios yields a point that falls outside the feasible region. Curvature checks reveal that the straight line connecting two good points may cut through a zone of negative allocations.

This finding serves as a warning for automated strategies. Robo-advisors often steer investors toward an even portfolio, believing it minimizes risk. But if they encounter such curvature along the way, the advisor subtly suggests going into debt. Stochastic market processes don’t forgive mistakes, and boundary equations expose the invisible barriers.

Fortunately, if the combining rule is concave (probability measure behaves “calmly”), then the web of connections inside the triangle simplifies and the center is safe again. But in general, the straight road to the middle doesn’t always lead to stability. The dartboard plays by its own rules.