On Minkowski's inequality and its application (original) (raw)
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Advances in Mathematics, 2014
Elementary proofs of sharp isoperimetric inequalities on a normed space (R n , •) equipped with a measure µ = w(x)dx so that w p is homogeneous are provided, along with a characterization of the corresponding equality cases. When p ∈ (0, ∞] and in addition w p is assumed concave, the result is an immediate corollary of the Borell-Brascamp-Lieb extension of the classical Brunn-Minkowski inequality, providing a new elementary proof of a recent result of Cabré-Ros Oton-Serra. When p ∈ (−1/n, 0), the relevant property turns out to be a novel "q-complemented Brunn-Minkowski" inequality: ∀λ ∈ (0, 1) ∀ Borel sets A, B ⊂ R n such that µ(R n \ A), µ(R n \ B) < ∞ , µ * (R n \ (λA + (1 − λ)B)) ≤ (λµ(R n \ A) q + (1 − λ)µ(R n \ B) q) 1/q , which we show is always satisfied by µ when w p is homogeneous with 1 q = 1 p + n; in particular, this is satisfied by the Lebesgue measure with q = 1/n. This gives rise to a new class of measures, which are "complemented" analogues of the class of convex measures introduced by Borell, but which have vastly different properties. The resulting isoperimetric inequality and characterization of isoperimetric minimizers extends beyond the recent results of Cañete-Rosales and Howe. The isoperimetric and Brunn-Minkowski type inequalities also extend to the non-homogeneous setting, under a certain log-convexity assumption on the density. Finally, we obtain functional, Sobolev and Nash-type versions of the studied inequalities.
The Converse of the Minkowski's Inequality Theorem and its Generalization
Proceedings of the American Mathematical Society, 1990
Let (£2, X, p.) be a measure space with two sets A, B el. such that 0 < p(A) < 1 < p(B) < oo , and let ?>:R+ -+ R+ be bijective and (¡> continuous at 0. We prove that if for all //-integrable step functions JCy:fi->R, </>~X (I <po\x+y\dp) <y~X (j <po\x\dp) +<p~* ( tpo\y\dpAj then tp(t) = <p{l)tp for some p > 1 . In the case of normalized measure we prove a generalization of Minkowski's inequality theorem. The suitable results for the reversed inequality are also presented.
The converse theorem for Minkowski's inequality
Indagationes Mathematicae, 2004
Let (s2, ~, #) be a measure space and ~, ~ : (0, co) ~ (0, ec) some bijective functions. Suppose that the functional P~,~ defined on class of #-integrable simple functions x : S? --, [0, ec), #({w : x(w) > 0} > 0, by the formula satisfies the triangle inequality. We prove that if there are A, B C C such that 0 < #(A) < 1 </~(B) < ee, the function ~ o ~p is superadditive, and limit0 ~(t) = 0 then there is ap > 1 such that
The Reverse-log-Brunn-Minkowski inequality
arXiv (Cornell University), 2023
Firstly, we propose our conjectured Reverse-log-Brunn-Minkowski inequality (RLBM). Secondly, we show that the (RLBM) conjecture is equivalent to the log-Brunn-Minkowski (LBM) conjecture proposed by Böröczky-Lutwak-Yang-Zhang. We name this as "reverse-toforward principle". Using this principle, we give a very simple new proof of the log-Brunn-Minkowski inequality in dimension two. Finally, we establish the "reverse-to-forward principle" for the log-Minkowski inequality (LM). Using this principle, we prove the log-Minkowski inequality in the case that one convex body is a zonoid (the inequality part was first proved by van Handel). Via a study of the lemma of relations, the full equality conditions ("dilated direct summands") are also characterized, which turns to be new.
Minkowski’s Integral Inequality for Function Norms
Birkhäuser Basel eBooks, 1995
Let ρ and λ be Banach function norms with the Fatou property. Then the generalized Minkowski integral inequality ρ(λ(f x)) ≤ M λ(ρ(f y)) holds for all measurable functions f (x, y) and some fixed constant M if and only if there exists 1 ≤ p ≤ ∞ such that λ is p-concave and ρ is p-convex.
New Minkowski and related inequalities via general kernels and measures
Journal of Inequalities and Applications
In this article, we introduce a class of functions mathfrakU(mathfrakp)\mathfrak{U}(\mathfrak{p})mathfrakU(mathfrakp) U ( p ) with integral representation defined over a measure space with σ-finite measure. The main purpose of this paper is to extend the Minkowski and related inequalities by considering general kernels. As a consequence of our general results, we connect our results with various variants for the fractional integrals operators. Such applications have wide use and importance in the field of applied sciences.