The Simplex Gravitational Illusion: A Repulsive Center ✛ ✎

Look at an equilateral triangle — the probability simplex for three assets. Each interior point is a portfolio: non-negative shares summing to one. The triangle's center, the barycenter, corresponds to an equal-weight allocation (1/3, 1/3, 1/3). It seems that by moving toward this point, you lower risk and avoid dangerous edges. But under certain conditions, the path to the center leads through invisible chasms.

The geometric trap works like this: the aggregation function η_ψ, which defines the portfolio map in stochastic portfolio theory, warps the admissible region. If the curvature of this function at the barycenter is positive, then there exist pairs of perfectly valid portfolios for which their midpoint, the half-sum, already violates the short-sale constraint. In other words, the region ceases to be convex, and the simplex itself becomes a field with gravitational anomalies.

The key to the puzzle is the normal form of constraints, derived through a coordinate-aggregation decomposition:

\[ $G_i$(\mu) = B + \psi'(\mu_i) + \sum_{j=1}^n \eta_\psi(\mu_j) \]

The term ψ'($μ_i$) accounts for the contribution of the coordinate itself, while the sum η_ψ($μ_j$) accumulates the influence of all assets. It is precisely the nonlinearity of the aggregation function that turns the center into a gravitational funnel. The measure of this nonlinearity is the Jensen difference — an analogue of a convexity test in the calculus of variations:

\[ $J_{\eta_\psi}$(x,h) = \eta_\psi(x+h) + \eta_\psi(x-h) - 2\eta_\psi(x) \]

If J is positive for small h, the function 'bulges' upward, and portfolio averaging produces an unexpected pitfall. In the language of probability theory, this means that conditional expectation breaks free of linear approximation.

This effect is not just a curiosity. It strikes at the popular heuristic 'equal-weight diversification minimizes short-sale risk,' embedded in many robo-advisory algorithms. When a robot advises moving toward the average market, it may unwittingly recommend portfolios with negative weights: that is, debts. Analysis of portfolio strategies and dynamical mappings must account for such geometry, and the theory of simplex graphs helps visualize these connections. Reassuring sufficient conditions have been found: if η_ψ is concave or its second derivative strictly dominates, the 'center' becomes a safe haven again. Interestingly, the entropy case (logarithmic function) restores the simplest geometry — the region is defined by independent thresholds.

Think back to the triangle picture. At first, the center seems to glow with safety. But now you know: sometimes that light isn't a beacon, it's lava in the crater.