Symmetry Results for Overdetermined Problems via $$L_p$$ Brunn–Minkowski Inequalities

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.

In 1999, Dar conjectured that there is a stronger version of the celebrated Brunn-Minkowski inequality. However, as pointed out by Campi, Gardner, and Gronchi in 2011, this problem seems to be open even for planar $o$-symmetric convex bodies. In this paper, we give a positive answer to Dar’s conjecture for all planar convex bodies. We also give the equality condition of this stronger inequality. For planar $o$-symmetric convex bodies, the log–Brunn–Minkowski inequality was established by Boroczky, Lutwak, Yang, and Zhang in 2012. It is stronger than the classical Brunn–Minkowski inequality, for planar $o$-symmetric convex bodies. Gaoyong Zhang asked if there is a general version of this inequality. Fortunately, the solution of Dar’s conjecture, especially, the definition of “dilation position”, inspires us to obtain a general version of the log–Brunn–Minkowski inequality. As expected, this inequality implies the classical Brunn–Minkowski inequality for all planar convex bodies.

The L p L^p -Brunn–Minkowski theory for p ≥ 1 p\geq 1 , proposed by Firey and developed by Lutwak in the 90’s, replaces the Minkowski addition of convex sets by its L p L^p counterpart, in which the support functions are added in L p L^p -norm. Recently, Böröczky, Lutwak, Yang and Zhang have proposed to extend this theory further to encompass the range p ∈ [ 0 , 1 ) p \in [0,1) . In particular, they conjectured an L p L^p -Brunn–Minkowski inequality for origin-symmetric convex bodies in that range, which constitutes a strengthening of the classical Brunn-Minkowski inequality. Our main result confirms this conjecture locally for all (smooth) origin-symmetric convex bodies in R n \mathbb {R}^n and p ∈ [ 1 − c n 3 / 2 , 1 ) p \in [1 - \frac {c}{n^{3/2}},1) . In addition, we confirm the local log-Brunn–Minkowski conjecture (the case p = 0 p=0 ) for small-enough C 2 C^2 -perturbations of the unit-ball of ℓ q n \ell _q^n for q ≥ 2 q \geq 2 , when the dimension n n is sufficiently large, as well as for the cube, which we show is the conjectural extremal case. For unit-balls of ℓ q n \ell _q^n with q ∈ [ 1 , 2 ) q \in [1,2) , we confirm an analogous result for p = c ∈ ( 0 , 1 ) p=c \in (0,1) , a universal constant. It turns out that the local version of these conjectures is equivalent to a minimization problem for a spectral-gap parameter associated with a certain differential operator, introduced by Hilbert (under different normalization) in his proof of the Brunn–Minkowski inequality. As applications, we obtain local uniqueness results in the even L p L^p -Minkowski problem, as well as improved stability estimates in the Brunn–Minkowski and anisotropic isoperimetric inequalities.

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

The Brunn–Minkowski inequality for volume differences

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

Let $(C_m^d)_{\infty}$ denote the graph whose set of vertices is $Z_m^d$ in which two distinct vertices are adjacent iff in each coordinate either they are equal or they differ, modulo m, by at most 1. Bollobás, Kindler, Leader, and O'Donnell proved that the minimum possible cardinality of a set of vertices of $(C_m^d)_{\infty}$ whose deletion destroys all topologically nontrivial cycles is $m^d-(m-1)^d$. We present a short proof of this result, using the Brunn–Minkowski inequality, and also show that the bound can be achieved only by selecting a value $x_i$ in each coordinate i, $1\leq i\leq d$, and by keeping only the vertices whose ith coordinate is not $x_i$ for all i.

A (one-dimensional) free Brunn–Minkowski inequality

Suppose that Ω and Ω1 are convex, open subsets of Rn. Denote their convex combination by The Brunn-Minkowski inequality says that (vol Ω)t≥ (1 -t) vol Ω0 1/N +t Vol Ω for 0≤t ≤ l. Moreover, if there is equality for some t other than an endpoint, then the domains Ω1 and Ω0 are translates and dilates of each other. Borell proved an analogue of the Brunn—Minkowski inequality with capacity (defined below) in place of volume. Borel’s theorem [B] says THEOREM A. Let Ωt= tΩ1+ (1—t)Ω0 be a convex combination of two convex subsets of RN,N≥3. Then cap The main purpose of this note is to prove.

We further consider the Orlicz dual Brunn-Minkowski theory. An Orlicz radial harmonic addition is introduced, which generalizes the Lp-radial addition and the Lp-harmonic addition to an Orlicz space, respectively. The variational formula for the dual mixed quermassintegrals with respect to the Orlicz radial harmonic addition is proved, and the new Orlicz dual quermassintegrals generalizes the Lp-dual quermassintegrals. The fundamental notions and conclusions of the dual quermassintegrals and the Minkoswki and Brunn-Minkowski inequalities for the dual quermassintegrals are extended to an Orlicz setting. The new Orlicz-Minkowski and Brunn-Minkowski inequalities in special case yield the Orlicz dual Minkowski inequality and Orlicz dual Brunn-Minkowski inequality, which also imply the Lp-dual Minkowski inequality and Lp-dual Brunn-Minkowski inequality for the dual quermassintegrals. As application, a dual log-Minkowski inequality is proved.

Complemented Brunn–Minkowski inequalities and isoperimetry for homogeneous and non-homogeneous measures

On the analogue of the concavity of entropy power in the Brunn–Minkowski theory

We consider a functional $\mathcal{F}$ on the space of convex bodies in ℝ n of the form $$ {\mathcal{F}}(K)=\int_{\mathbb{S}^{n-1}} f(u) \mathrm{S}_{n-1}(K,du), $$ where $f\in C(\mathbb{S}^{n-1})$ is a given continuous function on the unit sphere of ℝ n , K is a convex body in ℝ n , n≥3, and S n−1(K,⋅) is the area measure of K. We prove that $\mathcal{F}$ satisfies an inequality of Brunn–Minkowski type if and only if f is the support function of a convex body, i.e., $\mathcal{F}$ is a mixed volume. As a consequence, we obtain a characterization of translation invariant, continuous valuations which are homogeneous of degree n−1 and satisfy a Brunn–Minkowski type inequality.

An approach to interpolation of compact subsets of {{mathbb {C}}}^n, including Brunn–Minkowski type inequalities for the capacities of the interpolating sets, was developed in [8] by means of plurisubharmonic geodesics between relative extremal functions of the given sets. Here we show that a much better control can be achieved by means of the geodesics between weighted relative extremal functions. In particular, we establish convexity properties of the capacities that are stronger than those given by the Brunn–Minkowski inequalities.