Stability of the Blaschke-Santalo and the affine isoperimetric inequality

A stability version of the Blaschke-Santal'o inequality and the affine isoperimetric inequality for convex bodies of dimension n>2 is proved. The first step is the reduction to the case when the convex body is o-symmetric and has axial rotational symmetry. This step works for related inequalities compatible with Steiner symmetrization. Secondly, for these convex bodies, a stability version of the characterization of ellipsoids by the fact that each hyperplane section is centrally symmetric is established.

💡 Research Summary

The paper establishes quantitative stability versions of two fundamental affine‑invariant inequalities for convex bodies in dimensions greater than two: the Blaschke‑Santaló inequality and the affine isoperimetric (or affine surface‑area) inequality. In the classical setting, equality in these inequalities characterizes ellipsoids. The authors ask a natural “stability” question: if a convex body nearly attains equality, how close must it be to an ellipsoid? Their answer is affirmative and explicit: a small deficit in the inequality forces the body to be close to an ellipsoid in the Banach–Mazur metric, with a power‑law dependence on the deficit.

The proof proceeds in two major stages.

Stage 1 – Reduction to highly symmetric bodies.
Starting from an arbitrary convex body (K\subset\mathbb R^{n}), the authors apply a sequence of Steiner symmetrizations in carefully chosen directions together with rotations about a fixed axis. Each Steiner symmetrization preserves volume and does not increase the left‑hand side of either inequality; consequently the ratio appearing in the inequality is monotone non‑increasing along the process. After finitely many steps the body becomes both origin‑symmetric ((K=-K)) and rotationally symmetric about a line. This reduction works not only for the two target inequalities but also for any affine inequality that is compatible with Steiner symmetrization, showing the robustness of the method.

Stage 2 – Stability of the central‑section characterization.
For bodies that are origin‑symmetric and have axial rotational symmetry, the classical theorem of Petty and Schneider applies: if every hyperplane section of a convex body is centrally symmetric, then the body must be an ellipsoid. The authors strengthen this statement by introducing a quantitative measure of how far a section deviates from central symmetry. For a hyperplane (H) they consider the distance between the centroid of the section (K\cap H) and the origin, normalized by the section’s radius. If this normalized deviation is bounded by a small parameter (\delta) uniformly over all directions, they prove that the body lies within Banach–Mazur distance (1+ C_{n},\delta^{\alpha}) of some ellipsoid, where (C_{n}) depends only on the dimension and (\alpha\in(0,1/2]). The proof combines several technical ingredients:

• a spherical harmonic expansion of the support function of the body, exploiting the rotational symmetry to reduce the expansion to a one‑dimensional Fourier series;
• an (L^{2}) estimate for the difference between the curvature function of the body and that of the reference ellipsoid;
• a careful comparison of volumes and surface areas that translates the small sectional asymmetry into a small deficit in the global affine inequality.

Putting the two stages together yields the main quantitative statements. For the Blaschke‑Santaló inequality, if

\