Abstract
We prove that the Lebesgue space of variable exponent L-p(center dot)(Omega) is modularly uniformly convex in every direction provided the exponent p is finite a.e. and different from 1 a.e. The notion of uniform convexity in every direction was first introduced by Garkavi for the case of a norm. The contribution made in this work lies in the discovery of a modular, uniform-convexity-like structure of L-p(center dot)(Omega), which holds even when the behavior of the exponent p(center dot) precludes uniform convexity of the Luxembourg norm. Specifically, we show that the modular rho(u) = integral(Omega)vertical bar u(x)vertical bar dx possesses a uniform-convexity-like structure even if the variable exponent is not bounded away from 1 or infinity. Our result is new and we present an application to fixed point theory.