| Résume | The (B)-theorem of Cordero-Erausquin, Fradelizi and Maurey states that if \gamma_n denotes the standard Gaussian in dimension n, K is an origin-symmetric convex set, one has for any t that \gamma_n(e^t K)2 ≥ \gamma_n(K) γ\gamma_n(e^{2t} K). Herscovici, Livshyts, Rotem and Volberg proved a stability version of this result, showing that if one has equality up to a factor (1 + δ) then the inradius of K must be either "very large" or "very small", where the bounds depend on δ and on n. We will present a new stability estimate which is dimension-free and also yields more precise information about bodies which are near-optimizers: every principal component of the covariance matrix of the measure obtained by restricting the Gaussian to K must either be at least 1 - O(√δ) or at most O(δ). Our method extends immediately to yield stability estimates for generalizations of the (B)-theorem, namely the "strong" and "functional" versions, which reduce to spectral questions about 1-log-concave measures on ℝn. Time permitting, we will also mention other stability estimates of similar flavor which we can obtain for such measures.
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