$L_1$ contains every two-dimensional normed space

David Yost

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Displaying similar documents to “ $L_{1}$ contains every two-dimensional normed space”

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Let X be a normed space and $G_{F} (X)$ the group of all linear surjective isometries of X that are finite-dimensional perturbations of the identity. We prove that if $G_{F} (X)$ acts transitively on the unit sphere then X must be an inner product space.

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Gennadiy Averkov, Horst Martini (2009)

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Let $ℳ^{d}$ be a d-dimensional normed space with norm ||·|| and let B be the unit ball in $ℳ^{d}$ . Let us fix a Lebesgue measure $V_{B}$ in $ℳ^{d}$ with $V_{B} (B) = 1$ . This measure will play the role of the volume in $ℳ^{d}$ . We consider an arbitrary simplex T in $ℳ^{d}$ with prescribed edge lengths. For the case d = 2, sharp upper and lower bounds of $V_{B} (T)$ are determined. For d ≥ 3 it is noticed that the tight lower bound of $V_{B} (T)$ is zero.

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Gh. Constantin (1973)

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Generalized versions of Ilmanen lemma: Insertion of $C^{1, ω}$ or $C_{loc}^{1, ω}$ functions

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We prove that for a normed linear space $X$ , if $f_{1} : X \to ℝ$ is continuous and semiconvex with modulus $ω$ , $f_{2} : X \to ℝ$ is continuous and semiconcave with modulus $ω$ and $f_{1} \leq f_{2}$ , then there exists $f \in C^{1, ω} (X)$ such that $f_{1} \leq f \leq f_{2}$ . Using this result we prove a generalization of Ilmanen lemma (which deals with the case $ω (t) = t$ ) to the case of an arbitrary nontrivial modulus $ω$ . This generalization (where a $C_{l o c}^{1, ω}$ function is inserted) gives a positive answer to a problem formulated by A. Fathi and M. Zavidovique in 2010.

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A generalized Kahane-Khinchin inequality

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The inequality $ʃ l o g | \sum a_{n} e^{2 π i φ_{n}} | d φ_{1} \dots d φ_{n} \geq C l o g (\sum | a_{n} {|^{2})}^{1 / 2}$ with an absolute constant C, and similar ones, are extended to the case of $a_{n}$ belonging to an arbitrary normed space X and an arbitrary compact group of unitary operators on X instead of the operators of multiplication by $e^{2 π i φ}$ .

A Marchaud type inequality

Jorge Bustamante (2022)

Commentationes Mathematicae Universitatis Carolinae

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Tord Sjödin (2018)

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Let $F$ be a closed subset of $ℝ^{n}$ and let $P (x)$ denote the metric projection (closest point mapping) of $x \in ℝ^{n}$ onto $F$ in $l^{p}$ -norm. A classical result of Asplund states that $P$ is (Fréchet) differentiable almost everywhere (a.e.) in $ℝ^{n}$ in the Euclidean case $p = 2$ . We consider the case $2 < p < \infty$ and prove that the $i$ th component $P_{i} (x)$ of $P (x)$ is differentiable a.e. if $P_{i} (x) \neq x_{i}$ and satisfies Hölder condition of order $1 / (p - 1)$ if $P_{i} (x) = x_{i}$ .

Displaying similar documents to “ L 1 contains every two-dimensional normed space”

Displaying similar documents to “ $L_{1}$ contains every two-dimensional normed space”