Journal of Convex Analysis 22 (2015), No. 1, 117--144
Copyright Heldermann Verlag 2015
Uniform Convexity of Paranormed Generalizations of Lp Spaces
Justyna Jarczyk
Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Góra, Szafrana 4a, 65-516 Zielona Góra, Poland
j.jarczyk@wmie.uz.zgora.pl
Janusz Matkowski
Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Góra, Szafrana 4a, 65-516 Zielona Góra, Poland
j.matkowski@wmie.uz.zgora.pl
[Abstract-pdf]
For a measure space $(\Omega ,\Sigma ,\mu )$ and a bijective increasing function $\varphi :\left[ 0,\infty \right) \rightarrow \left[0,\infty \right)$ the $L^{p}$-like paranormed ($F$-normed) function space with the paranorm of the form $\mathbf{p}_{\varphi }(x)=\varphi ^{-1}\left( \int_{\Omega }\varphi \circ \left\vert x\right\vert d\mu \right)$ is considered. Main results give general conditions under which this space is uniformly convex. The Clarkson theorem on the uniform convexity of $L^{p}$-space is generalized. Under some specific assumptions imposed on $\varphi$ we give not only a proof of the uniform convexity but also show the formula of a modulus of convexity. We establish the uniform convexity of all finite-dimensional paranormed spaces, generated by a strictly convex bijection $\varphi$ of $[0, \infty)$. However, the {\it a contrario} proof of this fact provides no information on a modulus of convexity of these spaces. In some cases it can be done, even an exact formula of a modulus can be proved. We show how to make it in the case when $S={\mathbb R}^2$ and $\varphi$ is given by $\varphi(t)={\rm e}^t-1$.
Keywords: Lp-like paranorm, paranormed space, uniformly convex paranormed space, modulus of convexity, convex function, geometrically convex function, superquadratic function.
MSC: 46A16, 46E30; 47H09, 47H10
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