Local 𝐿^{𝑝}-Brunn–Minkowski inequalities for 𝑝<1

Given one metric measure space X satisfying a linear Brunn–Minkowski inequality, and a second one Y satisfying a Brunn–Minkowski inequality with exponent p≥−1, we prove that the product X×Y with the standard product distance and measure satisfies a Brunn–Minkowski inequality of order 1/(1+p−1) under mild conditions on the measures and the assumption that the distances are strictly intrinsic. The same result holds when we consider restricted classes of sets. We also prove that a linear Brunn–Minkowski inequality is obtained in X×Y when Y satisfies a Prékopa–Leindler inequality.In particular, we show that the classical Brunn–Minkowski inequality holds for any pair of weakly unconditional sets in Rn (i.e., those containing the projection of every point in the set onto every coordinate subspace) when we consider the standard distance and the product measure of n one-dimensional real measures with positively decreasing densities. This yields an improvement of the class of sets satisfying the Gaussian Brunn–Minkowski inequality.Furthermore, associated isoperimetric inequalities as well as recently obtained Brunn–Minkowski's inequalities are derived from our results.

In the paper, our main aim is to generalize the dual affine quermassintegrals to the Orlicz space. Under the framework of Orlicz dual Brunn–Minkowski theory, we introduce a new affine geometric quantity by calculating the first-order variation of the dual affine quermassintegrals, and call it the Orlicz dual affine quermassintegral. The fundamental notions and conclusions of the dual affine quermassintegrals and the Minkoswki and Brunn–Minkowski inequalities for them are extended to an Orlicz setting, and the related concepts and inequalities of Orlicz dual mixed volumes are also included in our conclusions. The new Orlicz–Minkowski and Orlicz–Brunn–Minkowski inequalities in a special case yield the Orlicz dual Minkowski inequality and Orlicz dual Brunn–Minkowski inequality, which also imply the L p {L_{p}} -dual Minkowski inequality and Brunn–Minkowski inequality for the dual affine quermassintegrals.

The Brunn–Minkowski Inequality, Minkowski's First Inequality, and Their Duals

Recently Boroczky, Lutwak, Yang and Zhang have proved the log-Brunn–Minkowski inequality which is stronger than the classical Brunn–Minkowski inequality for two origin-symmetric convex bodies in the plane. This paper presents a new proof of this inequality and proves the uniqueness of the cone-volume measure by using the log-Minkowski inequality.

The hyperbolic space $${\mathbb{H}^d}$$ can be defined as a pseudo-sphere in the (d + 1) Minkowski space-time. In this paper, a Fuchsian group Γ is a group of linear isometries of the Minkowski space such that $${\mathbb{H}^d/\Gamma}$$ is a compact manifold. We introduce Fuchsian convex bodies, which are closed convex sets in Minkowski space, globally invariant for the action of a Fuchsian group. A volume can be associated to each Fuchsian convex body, and, if the group is fixed, Minkowski addition behaves well. Then Fuchsian convex bodies can be studied in the same manner as convex bodies of Euclidean space in the classical Brunn–Minkowski theory. For example, support functions can be defined, as functions on a compact hyperbolic manifold instead of the sphere. The main result is the convexity of the associated volume (it is log concave in the classical setting). This implies analogs of Alexandrov–Fenchel and Brunn–Minkowski inequalities. Here the inequalities are reversed.

A new concept of p- Aleksandrov body is firstly introduced. In this paper, p- Brunn-Minkowski inequality and p- Minkowski inequality on the p- Aleksandrov body are established. Furthermore, some pertinent results concerning the Aleksandrov body and the p- Aleksandrov body are presented. 2000 Mathematics Subject Classification: 52A20 52A40

The Brunn–Minkowski inequality for volume differences

The dual Orlicz–Brunn–Minkowski theory

The Orlicz-Brunn-Minkowski theory, introduced by Lutwak, Yang, and Zhang, is a new extension of the classical Brunn-Minkowski theory. It represents a generalization of the $L_p$-Brunn-Minkowski theory, analogous to the way that Orlicz spaces generalize $L_p$ spaces. For appropriate convex functions $\varphi : [0,\infty)^m \to [0,\infty)$, a new way of combining arbitrary sets in $\mathbb{R}^n$ is introduced. This operation, called Orlicz addition and denoted by ${+}_{\varphi}$, has several desirable properties, but is not associative unless it reduces to $L_p$ addition. A general framework is introduced for the Orlicz-Brunn-Minkowski theory that includes both the new addition and previously introduced concepts, and makes clear for the first time the relation to Orlicz spaces and norms. It is also shown that Orlicz addition is intimately related to a natural and fundamental generalization of Minkowski addition called $M$-addition. The results obtained show, roughly speaking, that the Orlicz-Brunn-Minkowski theory is the most general possible based on an addition that retains all the basic geometrical properties enjoyed by the $L_p$-Brunn-Minkowski theory. Inequalities of the Brunn-Minkowski type are obtained, both for M-addition and Orlicz addition. The new Orlicz-Brunn-Minkowski inequality implies the $L_p$-Brunn-Minkowski inequality. New Orlicz-Minkowski inequalities are obtained that generalize the $L_p$-Minkowski inequality. One of these has connections with the conjectured log-Brunn-Minkowski inequality of Lutwak, Yang, and Zhang, and in fact these two inequalities together are shown to split the classical Brunn-Minkowski inequality.

We interpret the log-Brunn–Minkowski conjecture of Böröczky–Lutwak–Yang–Zhang as a spectral problem in centro-affine differential geometry. In particular, we show that the Hilbert–Brunn–Minkowski operator coincides with the centro-affine Laplacian, thus obtaining a new avenue for tackling the conjecture using insights from affine differential geometry. As every strongly convex hypersurface in \mathbb{R}^n is a centro-affine unit sphere, it has constant centro-affine Ricci curvature equal to n-2 , in stark contrast to the standard weighted Ricci curvature of the associated metric-measure space, which will in general be negative. In particular, we may use the classical argument of Lichnerowicz and a centro-affine Bochner formula to give a new proof of the Brunn–Minkowski inequality. For origin-symmetric convex bodies enjoying fairly generous curvature pinching bounds (improving with dimension), we are able to show global uniqueness in the L^p - and log-Minkowski problems, as well as the corresponding global L^p - and log-Minkowski conjectured inequalities. As a consequence, we resolve the isomorphic version of the log-Minkowski problem: for any origin-symmetric convex body \bar K in \mathbb{R}^n , there exists an origin-symmetric convex body K with \bar K \subset K \subset 8 \bar K such that K satisfies the log-Minkowski conjectured inequality, and such that K is uniquely determined by its cone-volume measure V_K . If \bar K is not extremely far from a Euclidean ball to begin with, an analogous isometric result, where 8 is replaced by 1+\varepsilon , is obtained as well.

Star Valuations and Dual Mixed Volumes

We formulate and prove a converse for a generalization of the classical Minkowski's inequality. The case when is also considered. Applying the same technique, we obtain an analog converse theorem for integral Minkowski's type inequality.

We give the counter-examples related to a Gaussian Brunn- Minkowski inequality and the (B) conjecture.

The entropy power inequality states that the effective variance (entropy power) of the sum of two independent random variables is greater than the sum of their effective variances. The Brunn-Minkowski inequality states that the effective radius of the set sum of two sets is greater than the sum of their effective radii. Both these inequalities are recast in a form that enhances their similarity. In spite of this similarity, there is as yet no common proof of the inequalities. Nevertheless, their intriguing similarity suggests that new results relating to entropies from known results in geometry and vice versa may be found. Two applications of this reasoning are presented. First, an isoperimetric inequality for entropy is proved that shows that the spherical normal distribution minimizes the trace of the Fisher information matrix given an entropy constraint--just as a sphere minimizes the surface area given a volume constraint. Second, a theorem involving the effective radii of growing convex sets is proved.

In the classical Brunn–Minkowski theory, a fruitful principle for obtaining new functionals is to apply familiar functionals, such as the volume $$V_n$$ , to a Minkowski linear combination $$K+\epsilon L$$ . This approach can also be used for Orlicz combinations. In this paper, applying the mixed volume functional to Orlicz combination, we introduced the Orlicz mixed volumes and proved the variational formula for the mixed volume with respect to the Orlicz combination. Furthermore, the Orlicz mixed Minkowski inequality and the Orlicz mixed Brunn–Minkowski inequality are established and the equivalence between the inequalities is demonstrated.