Upper Minkowski dimension estimates for convex restrictions (original) (raw)

Monotone and convex restrictions of continuous functions

Journal of Mathematical Analysis and Applications, 2017

Suppose that f belongs to a suitably defined complete metric space C α of Hölder α-functions defined on [0, 1]. We are interested in whether one can find large (in the sense of Hausdorff, or lower/upper Minkowski dimension) sets A ⊂ [0, 1] such that f | A is monotone, or convex/concave. Some of our results are about generic functions in C α like the following one: we prove that for a generic f ∈ C α 1 [0, 1], 0 < α < 2 for any A ⊂ [0, 1] such that f | A is convex, or concave we have dim H A ≤ dim M A ≤ max{0, α − 1}. On the other hand we also have some results about all functions belonging to a certain space. For example the previous result is complemented by the following one: for 1 < α ≤ 2 for any f ∈ C α [0, 1] there is always a set A ⊂ [0, 1] such that dim H A = α − 1 and f | A is convex, or concave on A.

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 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.

A generalization of Hölder and Minkowski inequalities

emis.ams.org

In this work, we give a generalization of Hölder and Minkowski inequalities to normal sequence algebras with absolutely monotone seminorm. Our main result is Theorem 2.1 and Theorem 2.2 which state these extensions. Taking F = 1 and · F = · 1 in both these theorems, we obtain classical versions of these inequalities. Also, using these generalizations we construct the vectorvalued sequence space F (X, λ, p) as a paranormed space which is a most general form of the space c 0 (X, λ, p) investigated in .

On Minkowski's inequality and its application

Journal of Inequalities and Applications, 2011

In the paper, we first give an improvement of Minkowski integral inequality. As an application, we get new Brunn-Minkowski-type inequalities for dual mixed volumes.

On the monotonicity of Minkowski sums towards convexity

2017

Let us define for a compact set A ⊂ R n the sequence A(k) = a 1 + • • • + a k k : a 1 ,. .. , a k ∈ A = 1 k A + • • • + A k times. By a theorem of Shapley, Folkman and Starr (1969), A(k) approaches the convex hull of A in Hausdorff distance as k goes to ∞. Bobkov, Madiman and Wang (2011) conjectured that Vol n (A(k)) is non-decreasing in k, where Vol n denotes the n-dimensional Lebesgue measure, or in other words, that when one has convergence in the Shapley-Folkman-Starr theorem in terms of a volume deficit, then this convergence is actually monotone. We prove that this conjecture holds true in dimension 1 but fails in dimension n ≥ 12. We also discuss some related inequalities for the volume of the Minkowski sum of compact sets, showing that this is fractionally superadditive but not supermodular in general, but is indeed supermodular when the sets are convex. Then we consider whether one can have monotonicity in the Shapley-Folkman-Starr theorem when measured using alternate measures of non-convexity, including the Hausdorff distance, effective standard deviation or inner radius, and a non-convexity index of Schneider. For these other measures, we present several positive results, including a strong monotonicity of Schneider's index in general dimension, and eventual monotonicity of the Hausdorff distance and effective standard deviation. Along the way, we clarify the interrelationships between these various notions of non-convexity, demonstrate applications of our results to combinatorial discrepancy theory, and suggest some questions worthy of further investigation.

On a characterization of LsppL\sp pLspp-norm and a converse of Minkowski's inequality

Hiroshima Mathematical Journal

Let C be a cone in a linear space. Under some weak regularity conditions we show that every subadditive function p: C -> R such that p(rx) < rp(x) for some re(0,1) and all xeC must be positively homogenous. As an application we obtain a new characterization of ZΛnorm. This permits us to prove among other things the following converse of Minkowski's inequality.

Minkowski’s theorem for arbitrary convex sets

European Journal of Combinatorics, 2008

V. Klee extended a well-known theorem of Minkowski to non-compact convex sets. We generalize Minkowski's theorem to convex sets which are not necessarily closed.

On Hölder's inequalities for convexity

Journal of Mathematical Analysis and Applications, 1989

This paper discusses Holder's inequalities obtained through the fundamental inequality of convexity (see [ 11) and presents some generalizations and complementary (or reverse) inequalities for Holder's inequalities. Correspondingly, the conclusion on the complementary inequalities for Hdlder's inequalities concerning finite series generalizes the result due to J. Aczel and E. F. Beckenbach [2]. Using the main result of this paper, we obtain some complementary inequalities for Minkowski's inequalities.