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We analyze the modular geometry of the Lebesgue space with variable exponent, Lp(·). Our central result is that Lp(·) possesses a modular uniform convexity property. Part of the novelty is that the property holds even in the case sup x∈Ω p(x) = ∞. We present specific applications to fixed point theory.

Modular Uniform Convexity of Lebesgue Spaces of Variable Integrability