Uniform Convexity of Paranormed Generalizations of L^p Spaces

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 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 $\varphi(t)={\rm e}^t-1$.