In two joint works with Neeman, we have resolved one of the most important aims of the project, pertaining to the multi-bubble isoperimetric conjectures on Euclidean space R^n and spherical space S^n. It was conjectured by J. Sullivan in the 1990's that standard bubbles uniquely minimize total perimeter among all k-bubbles enclosing prescribed volume, for any k≤n+1. These conjectures are known to hold when n=2, but the only other previously known case was the groundbreaking confirmation of the double-bubble conjecture (case k=2) in R^3 (and later R^n) circa 2000 by Hutchings-Morgan-Ritoré-Ros. We confirmed the conjectures on R^n and S^n in the following range: the double-bubble conjectures for n≥2, the triple-bubble conjectures (k=3) for n≥3, the quadruple-bubble conjectures (k=4) for n≥4 and the quintuple-bubble conjectures (k=5) for n≥5 (but without uniqueness on R^n). In fact, we showed that for all k≤n, a minimizing cluster necessarily has spherical interfaces, and that its cells (including the unbounded one) are necessarily connected, resolving a conjecture of Heppes. Furthermore, we showed that the corresponding Jacobi operator has exactly k negative eigenvalues, thereby establishing the concavity of the isoperimetric profile.
In a joint work with our post-doc Xu, we established that all standard bubbles on R^n, S^n and H^n (hyperbolic space) are stable, an infinitesimal local form of minimizing perimeter. Moreover, our results apply to Mobius-flat non-standard bubbles and even more general partitions as well.
We also studied the isoperimetric problem on the n-dimensional cube. The isoperimetric conjecture on the 3D cube predicts that minimizers are enclosed by spheres about a corner, cylinders about an edge and coordinate planes. This has only been previously established for relative volumes close to 0, 1/2 and 1 by compactness arguments. After decades without progress, our analysis confirms the conjecture on a cube with side lengths (β,1,1) in a new range of relatives volumes v∈[0,1/2]. In particular, we confirm the conjecture for the standard cube (β=1) for all v≤0.120582 when β≤0.919431 for the entire range where spheres are conjectured to be minimizing, and also for all v∈[0,1/2]∖(1/π−β/4,1/π+β/4). As for higher dimensions, we showed that, surprisingly, the analogous conjecture is false when n≥10.
In two joint works with Ivaki, we investigated refinements of the Brunn-Minkowski inequality, a key tool for understanding isoperimetric properties of convex bodies K in R^n. The log-Brunn-Minkowski conjecture predicts that when K is origin-symmetric, it enjoys improved volumetric properties, as captured by the cone-volume measure V_K. It was shown by Brendle-Choi-Daskalopoulos that if V_K is a multiple of the Lebesgue measure, then K is necessarily a centered Euclidean ball, thereby resolving a longstanding conjecture of Firey. Using a local version of the Brunn–Minkowski inequality and centro-affine differential geometry, we obtained a new proof of Firey’s conjecture (which generalizes to a multitude of additional nonlinearities), and moreover, we confirmed the log-Minkowski conjecture whenever K enjoys very liberal curvature pinching estimates, improving upon all previously known results.
In a joint work with Shabelman and Yehudayoff, we resolved a longstanding conjecture from the 1990s of Lutwak and Gardner regarding the intersection body operator I. Intersection bodies play a cardinal role in the dual Brunn-Minkowski theory, and so a natural question is to characterize the fixed and periodic points of I. We confirmed that IK = c K iff K is a centered Euclidean ball, and that I^2 K = c K iff K is a centered ellipsoid. To this end, we revisited Busemann's isoperimetric inequality for the volume of the intersection body and its monotonicity under Steiner symmetrization, characterizing the equality case.
In a solo work, we obtained a new proof of Royen's Gaussian Correlation Inequality using an inverse Brascamp-Lieb inequality for even functions which are more-log-concave than prescribed Gaussians, generalizing a recent inequality of Nakamura-Tsuji.
Our post-doc Bizeul obtained a new and simplified proof of the very recent groundbreaking confirmation of the Slicing Problem by Klartag-Lehec, which is intimately related to the KLS isoperimetric conjecture for convex bodies. Furthermore, Bizeul studied Gaussian-like isoperimetry in the form of the log-Sobolev inequality for general sub-Gaussian log-concave measures, which may be seen as a sub-Gaussian analogue of the KLS conjecture.
Our post-doc Xu and his collaborators established affine isoperimetric type inequalities for static convex domains in hyperbolic space, with equality being attained uniquely for hyperbolic ellipsoids. Furthermore, they established new Heintze-Karcher-type inequalities (a useful tool for establishing isoperimetric inequalities as shown by Gromov, Ros, and others) in sub-static warped product manifolds, and derived uniqueness results for a class of curvature equations.
