Minkowski Inequality Research Articles

The self-shrinker in warped product space and the weighted Minkowski inequality

This paper consists of two parts. One is that for a kind of self-shrinker in a manifold with warped product metric, we prove that under some conditions on ambient space, the mean convex self-shrinker must have parallel second fundamental form. The other one is a generalization of Brendle’s Minkowski inequality for weighted mean curvature.

Grünbaum's inequality for projections

Let K be a convex body in Rn whose centroid is at the origin, let E∈G(n,k) be a subspace, and let ξ∈Sn−1. We find the best constant c=(kn+1)k so that |(K|E)∩ξ+|k≥c|K|E|k, and completely determine the minimizer. Here, |⋅|k is k-dimensional volume, K|E is the projection of K onto E, and ξ+={x∈Rn:〈x,ξ〉≥0}. Our result generalizes both Grünbaum's inequality, and an old inequality of Minkowski and Radon.

Orlicz–Brunn–Minkowski inequalities for Blaschke–Minkowski homomorphisms

Orlicz–Brunn–Minkowski type inequalities for Blaschke–Minkowski homomorphisms and their polars are established.

Several inequalities for the pan-integral

Several inequalities for the pan-integral are investigated. It is shown that the Chebyshev inequality holds for an arbitrary subadditive measure if and only if the integrands f, g are comonotone. Thus, we provide a new characterization for nonnegative comonotone functions. It is also shown that the Hölder and Minkowski inequalities for the pan-integral hold if the monotone measure μ is subadditive. Since the pan-integral coincides with the concave integral when μ is subadditive, our results can also be applied to the concave integral.

The dual difference Aleksandrov–Fenchel inequality

Recently, a dual Minkowski inequality and a dual Brunn–Minkowski inequality for volume differences were established. Following this, in this paper we establish a dual Aleksandrov–Fenchel inequality for dual mixed volume differences which generalizes several recent results.

Geodesic interpolation inequalities on Heisenberg groups

In this Note, we present geodesic versions of the Borell–Brascamp–Lieb, Brunn–Minkowski and entropy inequalities on the Heisenberg group Hn. Our arguments use the Riemannian approximation of Hn combined with optimal mass-transportation techniques.

Generalizations of the Brunn–Minkowski inequality

Yuan and Leng (2007) gave a generalization of the matrix form of the Brunn–Minkowski inequality. In this note, we first give a simple proof of this inequality, and then show a generalization of this to a larger class of matrices, namely, matrices whose numerical ranges are contained in a sector.

Dar’s conjecture and the log–Brunn–Minkowski inequality

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.

A Curved Brunn Minkowski Inequality for the Symmetric Group

In this paper, we construct an injection $A \times B \rightarrow M \times M$ from the product of any two nonempty subsets of the symmetric group into the square of their midpoint set, where the metric is that corresponding to the conjugacy class of transpositions. If $A$ and $B$ are disjoint, our construction allows to inject two copies of $A \times B$ into $M \times M$. These injections imply a positively curved Brunn-Minkowski inequality for the symmetric group analogous to that obtained by Ollivier and Villani for the hypercube. However, while Ollivier and Villani's inequality is optimal, we believe that the curvature term in our inequality can be improved. We identify a hypothetical concentration inequality in the symmetric group and prove that it yields an optimally curved Brunn-Minkowski inequality.

Refinements of operator Cauchy–Schwarz and Minkowski inequalities for p-modified norms and related norm inequalities

For (degree) p-modifications Φ(p) of symmetrically norming functions Φ and finite families of bounded Hilbert space operators A1,B1,⋯,AN,BN and X‖∑n=1NAn⁎XBn‖Φ(p)≤‖(|∑n=1NAn⁎XBn|2+∑n=1N|An(‖∑n=1NAn⁎An‖12I+(‖∑n=1NAn⁎An‖I−∑n=1NAn⁎An)12)−1∑m=1NAm⁎XBm−‖∑n=1NAn⁎An‖12XBn|2)p2‖Φ1p=‖∑n=1NAn⁎An‖12‖(∑n=1NBn⁎X⁎XBn)12‖Φ(p) for 0<p≤2, while for p≥2‖∑n=1NAn⁎XBn‖Φ(p)≤‖|∑n=1NAn⁎XBn|p+∑n=1N|An(‖∑n=1NAn⁎An‖12I+(‖∑n=1NAn⁎An‖I−∑n=1NAn⁎An)12)−1∑m=1NAm⁎XBm−‖∑n=1NAn⁎An‖12XBn|p‖Φ1p≤‖∑n=1NAn⁎An‖12‖∑n=1NBn⁎X⁎XBn‖Φ(p2).For p≥2 this implies refined Minkowski inequality for p-modified unitarily invariant norms ‖⋅‖Φ(p)‖∑n=1NBn‖Φ(p)≤‖|∑n=1NBn|p+(∑n=1N‖Bn‖Φ(p))−p2∑n=1N‖Bn‖Φ(p)−p2|‖Bn‖Φ(p)∑m=1NBm−∑m=1N‖Bm‖Φ(p)Bn|p‖Φ1p≤∑n=1N‖Bn‖Φ(p).

The logarithmic Minkowski inequality for non-symmetric convex bodies

We validate the conjectured logarithmic Minkowski inequality, and thus the equivalent logarithmic Brunn–Minkowski inequality, in some particular cases and we prove some variants of the logarithmic Minkowski inequality for general convex bodies without the symmetry assumption. An application of one of these variants is shown.

Translative containment measure and symmetric mixed isohomothetic inequalities

We first investigate the translative containment measure for convex domain K0 to contain, or to be contained in, the homothetic copy of another convex domain K1, i.e., given two convex domains K0, K1 of areas A0, A1, respectively, in the Euclidean plane ℝ2, is there a translation T so that t(TK1) ⊂ K0 or t(TK1) ⊃ K0 for t > 0? Via the translative kinematic formulas of Poincare and Blaschke in integral geometry, we estimate the symmetric mixed isohomothetic deficit σ2(K0,K1) ≡ A012 − A0A1, where A01 is the mixed area of K0 and K1. We obtain a sufficient condition for K0 to contain, or to be contained in, t(TK1). We obtain some Bonnesen-style symmetric mixed isohomothetic inequalities and reverse Bonnesen-style symmetric mixed isohomothetic inequalities. These symmetric mixed isohomothetic inequalities obtained are known as Bonnesen-style isopermetric inequalities and reverse Bonnesen-style isopermetric inequalities if one of domains is a disc. As direct consequences, we obtain some inequalities that strengthen the known Minkowski inequality for mixed areas and the Bonnesen-Blaschke-Flanders inequality.

The generalized L p -mixed affine surface area

In this article, we put forward the concept of the (i, j)-type L p -mixed affine surface area, such that the notion of L p -affine surface area which be shown by Lutwak is its special cases. Furthermore, applying this concept, the Minkowski inequality for the (i,−p)-type L p -mixed affine surface area and the extensions of the well-known L p -Petty affine projection inequality are established, respectively. Besides, we give an affirmative answer for the generalized L p -Winterniz monotonicity problem.

Inequalities of quermassintegrals about mixed Blaschke Minkowski homomorphisms

In this article, we establish some inequalities of quermassintegrals associated with mixed Blaschke Minkowski homomorphisms. In particular, Minkowski and Brunn-Minkowski type inequalities for quermassintegrals differences of mixed Blaschke Minkowski homomorphisms are established. In addition, we also give an isolated form of Brunn-Minkowski type inequality of quermassintegrals established by Schuster.

Conditional Choquet Expectation

Capacity plays an important role in many areas. A capacity is usually studied under the assumption that it is submodular. In this article, we first present several inequalities for the submodular capacity and state some special submodular capacities. Second, we introduce the concept of conditional Choquet expectation which is the conditional expectation with respect to submodular capacity, and then we prove some properties of this concept, which include positive homogeneity, translation invariance, asymmetry, subadditivity, and monotonicity. Finally, we state that Cr inequality, Schwarz inequality, Hlder’s inequality, Minkowski inequality, and Jensen’s inequality still hold for this kind of conditional Choquet expectations.

A note on inequalities and critical values of fuzzy rough variables

A fuzzy rough variable is defined as a rough variable on the universal set of fuzzy variables, or a rough variable taking ‘fuzzy variable’ values. In order to further discuss the mathematical properties of fuzzy rough variables, this paper extends some inequalities to the context of fuzzy rough theory based on the chance measure and the expected value operator, involving the Markov inequality, the Chebyshev inequality, the Hölder inequality, the Minkowski inequality, and the Jensen inequality. After that, linearity, monotonicity, and continuity of critical values of fuzzy rough variables are also investigated.

Prove inequalities by solving maximum/minimum problems using a computer algebra system

This presentation offers an alternative to traditional approaches to proving non-trivial inequalities, such as applying AM-GM, Cauchy, Holder and Minkowski inequalities. This alternative approach, as demonstrated by various examples, establishes the validity of an inequality through solving a maximization/minimization problem by commonly practiced procedures in Calculus. Since the procedures are algorithmic, a Computer Algebra System (CAS) can carry out the computation efficiently.

On Xiang's Observations Concerning the Cauchy–Schwarz Inequality

We give a simple proof of an improved version of the Cauchy–Schwarz inequality for vectors and for functions, first observed by Xiang. Also we improve an inequality of Minkowski.

Improved [formula omitted]-mixed volume inequality for convex bodies

A sharp quantitative version of the Lp-mixed volume inequality is established. This is achieved by exploiting an improved Jensen inequality. This inequality is a generalization of Pinsker–Csiszár–Kullback inequality for the Tsallis entropy. Finally, a sharp quantitative version of the Lp-Brunn–Minkowski inequality is also proved as a corollary.

(K, N)-Convexity and the Curvature-Dimension Condition for Negative N

We extend the range of N to negative values in the (K, N)-convexity (in the sense of Erbar–Kuwada–Sturm), the weighted Ricci curvature $$\mathop {\mathrm {Ric}}\nolimits _N$$ and the curvature-dimension condition $$\mathop {\mathrm {CD}}\nolimits (K,N)$$ . We generalize a number of results in the case of $$N>0$$ to this setting, including Bochner’s inequality, the Brunn–Minkowski inequality and the equivalence between $$\mathop {\mathrm {Ric}}\nolimits _N \ge K$$ and $$\mathop {\mathrm {CD}}\nolimits (K,N)$$ . We also show an expansion bound for gradient flows of Lipschitz (K, N)-convex functions.