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On almost everywhere differentiability of the metric projection on closed sets in $l^{p} (ℝ^{n})$ , $2 < p < \infty$

Tord Sjödin (2018)

Czechoslovak Mathematical Journal

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 $P_{i} (x) = x_{i}$ .

On almost quasicontinuity

Anna Neubrunnová, Tibor Šalát (1992)

Mathematica Bohemica

The concept of almost quasicontinuity is investgated in this paper in several directions (e.g. the relation of this concept to other generalizations of continuity is described, various types of convergence of sequences of almost quasicontinuous function are studied, a.s.o.).

On alternative functional equations. (Short Communication).

Halina Swiatak (1976)

Aequationes mathematicae

On an Abstract Formulation of Regularity

Zdena Riečanová (1971)

Matematický časopis

On an application of hypoellipticity to solutions of functional equations.

Shigeru Haruki, Akira Tsutsumi (1996)

Aequationes mathematicae

On an elementary inclusion problem and generalized weighted quasi-arithmetic means

Zoltán Daróczy, Zsolt Páles (2013)

Banach Center Publications

The aim of this note is to characterize the real coefficients p₁,...,pₙ and q₁,...,qₖ so that $\sum_{i = 1}^{n} p_{i} x_{i} + \sum_{j = 1}^{k} q_{j} y_{j} \in c o n v x ₁, . . ., x ₙ$ be valid whenever the vectors x₁,...,xₙ, y₁,...,yₖ satisfy y₁,...,yₖ ⊆ convx₁,...,xₙ. Using this characterization, a class of generalized weighted quasi-arithmetic means is introduced and several open problems are formulated.