Radial p-th mean bodies are geometric objects constructed by averaging a convex body's radial function raised to the p-th power over all directions. Gardner and Zhang introduced them in 1998 and proved convexity for p ≥ 0, but the case -1 < p < 0 resisted proof for nearly three decades. The negative exponent inverts the averaging — instead of emphasizing large radial values, it emphasizes small ones — and the resulting geometry was not obviously convex.
The resolution uses Prékopa's theorem: if a function on R^(n+1) is log-concave, then its integral over one variable is log-concave. The connection to radial mean bodies comes through representing the body's gauge function as such an integral. The key step is showing that the integrand — involving the radial function raised to a negative power — maintains log-concavity despite the inversion, because the underlying body is convex and its radial function satisfies the right structural conditions.
As a byproduct, the proof provides a new derivation of Keith Ball's theorem on integrals of log-concave functions along rays and extends it to negative p-values. The Rogers-Shephard inequality and projection body theory both benefit from the generalization.
The 28-year gap between conjecture and proof is instructive. The obstacle wasn't the difficulty of the final proof — Prékopa's theorem has been available since the 1970s. The obstacle was recognizing which representation of the problem made the theorem applicable. The body needed to be viewed not as a geometric object to be proved convex, but as an integral to be shown log-concave. Same object, different framing, suddenly tractable.