The rearrangement inequalities of Hardy-Littlewood and Riesz say that certain integrals involving products of two or three functions increase under symmetric decreasing rearrangement. It is known that these inequalities extend to integrands of the form F(u_1,..., u_m) where F is supermodular; in particular, they hold when F has nonnegative mixed second derivatives. This paper concerns the regularity assumptions on F and the equality cases. It is shown here that extended Hardy-Littlewood and Riesz inequalities are valid for supermodular integrands that are just Borel measurable. Under some nondegeneracy conditions, all equality cases are equivalent to radially decreasing functions under transformations that leave the functionals invariant (i.e., measure-preserving maps for the Hardy-Littlewood inequality, translations for the Riesz inequality). The proofs rely on monotone changes of variables in the spirit of Sklar's theorem.
The Riesz inequality in Eq. ) is non-trivial even when @math is just a product of two functions. Ahlfors introduced two-point rearrangements to treat this case on @math @cite_22 , Baernstein-Taylor proved the corresponding result on @math @cite_19 , and Beckner noted that the proof remains valid on @math and @math @cite_9 . When @math is a product of @math functions, Eq. ) has applications to spectral invariants of heat kernels via the Trotter product formula @cite_15 . This case was settled by Friedberg-Luttinger @cite_13 , Burchard-Schmuckenschl "ager @cite_25 , and by Morpurgo, who proved Eq. ) more generally for integrands of the form with @math convex (Theorem 3.13 of @cite_8 ). In the above situations, equality cases have been determined @cite_26 @cite_21 @cite_25 @cite_8 . Almgren-Lieb used the technique of Crowe-Zweibel-Rosen -bloom to prove Eq. ) for @math @cite_5 . The special case where @math for some convex function @math was identified by Baernstein as a master inequality' from which many classical geometric inequalities can be derived quickly @cite_32 . Eq. ) for continuous supermodular integrands with @math is due to Draghici @cite_0 .
@cite_26 @cite_22 @cite_8 @cite_9 @cite_21 @cite_32 @cite_0 @cite_19 @cite_5 @cite_15 @cite_13 @cite_25
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A version of the Riesz-Sobolev convolution inequality is formulated and proved for arbitrary compact connected Abelian groups. Maximizers are characterized and a quantitative stability theorem is proved, under natural hypotheses. A corresponding stability theorem for sets whose sumset has nearly minimal measure is also proved, sharpening recent results of other authors. For the special case of the group @math , a continuous deformation of sets is developed, under which an appropriately scaled Riesz-Sobolev functional is shown to be nondecreasing. The full' edge isoperimetric inequality for the discrete cube (due to Harper, Bernstein, Lindsay and Hart) specifies the minimum size of the edge boundary @math of a set @math , as a function of @math . A weaker (but more widely-used) lower bound is @math , where equality holds iff @math is a subcube. In 2011, the first author obtained a sharp stability' version of the latter result, proving that if @math , then there exists a subcube @math such that @math . The weak' version of the edge isoperimetric inequality has the following well-known generalization for the @math -biased' measure @math on the discrete cube: if @math , or if @math and @math is monotone increasing, then @math . In this paper, we prove a sharp stability version of the latter result, which generalizes the aforementioned result of the first author. Namely, we prove that if @math , then there exists a subcube @math such that @math , where @math . This result is a central component in recent work of the authors proving sharp stability versions of a number of Erd o s-Ko-Rado type theorems in extremal combinatorics, including the seminal complete intersection theorem' of Ahlswede and Khachatrian. In addition, we prove a biased-measure analogue of the full' edge isoperimetric inequality, for monotone increasing sets, and we observe that such an analogue does not hold for arbitrary sets, hence answering a question of Kalai. We use this result to give a new proof of the full' edge isoperimetric inequality, one relying on the Kruskal-Katona theorem. In a celebrated work by Hoeffding [J. Amer. Statist. Assoc. 58 (1963) 13-30], several inequalities for tail probabilities of sums M n = X 1 + ... + X n of bounded independent random variables X j were proved. These inequalities had a considerable impact on the development of probability and statistics, and remained unimproved until 1995 when Talagrand [Inst. Hautes Etudes Sci. Publ. Math. 81 (1995a) 73-205] inserted certain missing factors in the bounds of two theorems. By similar factors, a third theorem was refined by Pinelis [Progress in Probability 43 (1998) 257-314] and refined (and extended) by me. In this article, I introduce a new type of inequality. Namely, I show that P M n ≥ x ≤ cP S n ≥ x , where c is an absolute constant and S n = e 1 + ... + e n is a sum of independent identically distributed Bernoulli random variables (a random variable is called Bernoulli if it assumes at most two values). The inequality holds for those x ∈ Ρ where the survival function x → P S n ≥ x has a jump down. For the remaining x the inequality still holds provided that the function between the adjacent jump points is interpolated linearly or log-linearly. If it is necessary, to estimate P S n ≥ x special bounds can be used for binomial probabilities. The results extend to martingales with bounded differences. It is apparent that Theorem 1.1 of this article is the most important. The inequalities have applications to measure concentration, leading to results of the type where, up to an absolute constant, the measure concentration is dominated by the concentration in a simplest appropriate model, such results will be considered elsewhere. By using optimal mass transportation and a quantitative Holder inequality, we provide estimates for the Borell–Brascamp–Lieb deficit on complete Riemannian manifolds. Accordingly, equality cases in Borell–Brascamp–Lieb inequalities (including Brunn–Minkowski and Prekopa–Leindler inequalities) are characterized in terms of the optimal transport map between suitable marginal probability measures. These results provide several qualitative applications both in the flat and non-flat frameworks. In particular, by using Caffarelli's regularity result for the Monge–Ampere equation, we give a new proof of Dubuc's characterization of the equality in Borell–Brascamp–Lieb inequalities in the Euclidean setting. When the n-dimensional Riemannian manifold has Ricci curvature Ric(M) ≥ (n-1)k for some k ⋲ ℝ, it turns out that equality in the Borell–Brascamp–Lieb inequality is expected only when a particular region of the manifold between the marginal supports has constant sectional curvature k. A precise characterization is provided for the equality in the Lott–Sturm–Villani-type distorted Brunn–Minkowski inequality on Riemannian manifolds. Related results for (not necessarily reversible) Finsler manifolds are also presented.