Bourgain’s Slicing Problem and KLS Isoperimetry up to Polylog
summary
Claim: Bourgain's slicing constant and KLS isoperimetric constant are bounded up to polylog factors — specifically L_n ≤ (log n)^4, σ_n ≤ (log n)^4 and ψ_n ≤ (log n)^5. Proof combines stochastic localization (Eldan), a functional H^{-1} inequality, heat-flow adjoints Q_s, spectral truncation and Poincaré estimates. The argument quantifies spectral mass decay under convolution and converts that into variance/isoperimetry bounds, yielding effective polylogarithmic control where only polynomial or exponential bounds existed before.
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