Hidden structures in the class of convex functions and a new duality transform

Shiri Artstein-Avidan; Vitali Milman

Displaying similar documents to “Hidden structures in the class of convex functions and a new duality transform”

Generalized characterization of the convex envelope of a function

Fethi Kadhi (2002)

RAIRO - Operations Research - Recherche Opérationnelle

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We investigate the minima of functionals of the form $\int_{[a, b]} g (\dot{u} (s)) d s$ where $g$ is strictly convex. The admissible functions $u : [a, b] \to ℝ$ are not necessarily convex and satisfy $u \leq f$ on $[a, b]$ , $u (a) = f (a)$ , $u (b) = f (b)$ , $f$ is a fixed function on $[a, b]$ . We show that the minimum is attained by $\bar{f}$ , the convex envelope of $f$ .

A sharp isoperimetric inequality in the plane

Angelo Alvino, Vincenzo Ferone, Carlo Nitsch (2011)

Journal of the European Mathematical Society

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We show that among all the convex bounded domain in $m a t h b b R^{2}$ having an assigned Fraenkel asymmetry index, there exists only one convex set (up to a similarity) which minimizes the isoperimetric deficit. We also show how to construct this set. The result can be read as a sharp improvement of the isoperimetric inequality for convex planar domain.

Smoothing a polyhedral convex function via cumulant transformation and homogenization

Alberto Seeger (1997)

Annales Polonici Mathematici

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Given a polyhedral convex function g: ℝⁿ → ℝ ∪ +∞, it is always possible to construct a family ${g ₜ}_{t > 0}$ which converges pointwise to g and such that each gₜ: ℝⁿ → ℝ is convex and infinitely often differentiable. The construction of such a family ${g ₜ}_{t > 0}$ involves the concept of cumulant transformation and a standard homogenization procedure.

On some results about convex functions of order $α$

M. Obradović, S. Owa (1986)

Matematički Vesnik

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The Young inequality and the Δ₂-condition

Philippe Laurençot (2002)

Colloquium Mathematicae

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If φ: [0,∞) → [0,∞) is a convex function with φ(0) = 0 and conjugate function φ*, the inequality $x y \leq ε φ (x) + C_{ε} φ * (y)$ is shown to hold true for every ε ∈ (0,∞) if and only if φ* satisfies the Δ₂-condition.

Poincaré Inequalities and Moment Maps

Bo’az Klartag (2013)

Annales de la faculté des sciences de Toulouse Mathématiques

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We discuss a method for obtaining Poincaré-type inequalities on arbitrary convex bodies in $ℝ^{n}$ . Our technique involves a dual version of Bochner’s formula and a certain moment map, and it also applies to some non-convex sets. In particular, we generalize the central limit theorem for convex bodies to a class of non-convex domains, including the unit balls of $ℓ_{p}$ -spaces in $ℝ^{n}$ for $0 < p < 1$ .

Convex universal fixers

Magdalena Lemańska, Rita Zuazua (2012)

Discussiones Mathematicae Graph Theory

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In [1] Burger and Mynhardt introduced the idea of universal fixers. Let G = (V, E) be a graph with n vertices and G’ a copy of G. For a bijective function π: V(G) → V(G’), define the prism πG of G as follows: V(πG) = V(G) ∪ V(G’) and $E (π G) = E (G) \cup E (G^{'}) \cup M_{π}$ , where $M_{π} = u π (u) | u \in V (G)$ . Let γ(G) be the domination number of G. If γ(πG) = γ(G) for any bijective function π, then G is called a universal fixer. In [9] it is conjectured that the only universal fixers are the edgeless graphs K̅ₙ. In this work we generalize the concept...

Convex integration with constraints and applications to phase transitions and partial differential equations

Stefan Müller, Vladimír Šverák (1999)

Journal of the European Mathematical Society

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We study solutions of first order partial differential relations $D u \in K$ , where $u : Ω \subset ℝ^{n} \to ℝ^{m}$ is a Lipschitz map and $K$ is a bounded set in $m \times n$ matrices, and extend Gromov’s theory of convex integration in two ways. First, we allow for additional constraints on the minors of $D u$ and second we replace Gromov’s $P$ −convex hull by the (functional) rank-one convex hull. The latter can be much larger than the former and this has important consequences for the existence of ‘wild’ solutions to elliptic systems. Our...

The Quantitative Isoperimetric Inequality for Planar Convex Domains

Carlo Nitsch (2008)

Bollettino dell'Unione Matematica Italiana

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We prove that among all the convex bounded domains in $\mathbb{R}^{2}$ having an assigned Fraenkel asymmetry index, there exists only one convex set (up to a similarity) which minimizes the isoperimetric deficit. We show how to construct this set. The result can be read as a sharp improvement of the isoperimetric inequality for convex planar domains.

On closed sets with convex projections in Hilbert space

Stoyu Barov, Jan J. Dijkstra (2007)

Fundamenta Mathematicae

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Let k be a fixed natural number. We show that if C is a closed and nonconvex set in Hilbert space such that the closures of the projections onto all k-hyperplanes (planes with codimension k) are convex and proper, then C must contain a closed copy of Hilbert space. In order to prove this result we introduce for convex closed sets B the set $^{k} (B)$ consisting of all points of B that are extremal with respect to projections onto k-hyperplanes. We prove that $^{k} (B)$ is precisely the intersection of...

A d.c. $C^{1}$ function need not be difference of convex $C^{1}$ functions

David Pavlica (2005)

Commentationes Mathematicae Universitatis Carolinae

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In [2] a delta convex function on $ℝ^{2}$ is constructed which is strictly differentiable at $0$ but it is not representable as a difference of two convex function of this property. We improve this result by constructing a delta convex function of class $C^{1} (ℝ^{2})$ which cannot be represented as a difference of two convex functions differentiable at 0. Further we give an example of a delta convex function differentiable everywhere which is not strictly differentiable at 0.

A "hidden" characterization of approximatively polyhedral convex sets in Banach spaces

Taras Banakh, Ivan Hetman (2012)

Studia Mathematica

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A closed convex subset C of a Banach space X is called approximatively polyhedral if for each ε > 0 there is a polyhedral (= intersection of finitely many closed half-spaces) convex set P ⊂ X at Hausdorff distance < ε from C. We characterize approximatively polyhedral convex sets in Banach spaces and apply the characterization to show that a connected component of the space $C o n v_{} (X)$ of closed convex subsets of X endowed with the Hausdorff metric is separable if and only if contains a...

The radius of close-to-convexity of $V_{k} (ϱ)$

Prakash G. Umarani (1983)

Annales Polonici Mathematici

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Convexity of sublevel sets of plurisubharmonic extremal functions

Finnur Lárusson, Patrice Lassere, Ragnar Sigurdsson (1998)

Annales Polonici Mathematici

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Let X be a convex domain in ℂⁿ and let E be a convex subset of X. The relative extremal function $u_{E, X}$ for E in X is the supremum of the class of plurisubharmonic functions v ≤ 0 on X with v ≤ -1 on E. We show that if E is either open or compact, then the sublevel sets of $u_{E, X}$ are convex. The proof uses the theory of envelopes of disc functionals and a new result on Blaschke products.

On convex and *-concave multifunctions

Bożena Piątek (2005)

Annales Polonici Mathematici

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A continuous multifunction F:[a,b] → clb(Y) is *-concave if and only if the inclusion $1 / (t - s) \int_{s}^{t} F (x) d x \subset (F (s) \frac{*}{+} F (t)) / 2$ holds for every s,t ∈ [a,b], s < t.

Graphs with convex domination number close to their order

Joanna Cyman, Magdalena Lemańska, Joanna Raczek (2006)

Discussiones Mathematicae Graph Theory

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For a connected graph G = (V,E), a set D ⊆ V(G) is a dominating set of G if every vertex in V(G)-D has at least one neighbour in D. The distance $d_{G} (u, v)$ between two vertices u and v is the length of a shortest (u-v) path in G. An (u-v) path of length $d_{G} (u, v)$ is called an (u-v)-geodesic. A set X ⊆ V(G) is convex in G if vertices from all (a-b)-geodesics belong to X for any two vertices a,b ∈ X. A set X is a convex dominating set if it is convex and dominating. The convex domination number $γ_{c o n} (G)$ of a...