Minkowski's problem asks: given a measure on the sphere describing how surface area is distributed across directions, what convex body has that surface area measure? The existence and uniqueness are classical. The stability question is harder: if two surface area measures are close, must the corresponding convex bodies be close?
Böröczky, Machado, and Ramos (arXiv:2603.17726) prove quantitative stability. If two probability measures on the sphere are close in the dual-convex distance d_C, then their Minkowski bodies have Fraenkel asymmetry bounded by α ≤ C · d_C^{(1+1/n)/2}, where n is the dimension. The exponent depends on dimension but is explicit.
The proof uses strong-concavity properties derived from quantitative Brunn-Minkowski and isoperimetric inequalities. The Minkowski body is characterized as the solution to a variational problem, and the stability estimate follows from the curvature of the variational functional near its maximum. Strong concavity means the functional curves sharply downward away from the optimizer, preventing nearby measures from producing distant bodies.
The result extends to L_p-Minkowski problems for 1 ≤ p ≠ n, where the surface area measure is replaced by a p-surface area measure that weights different directions differently. The stability estimates adapt, with the same proof strategy yielding analogous bounds in each case.
Stability of inverse problems is practically important. Reconstruction algorithms that recover a convex body from noisy measurements of its surface area need stability guarantees to ensure that small measurement errors don't produce wildly wrong reconstructions. The quantitative bound converts an abstract uniqueness theorem into a concrete error estimate: this much noise in the input produces at most this much error in the output.