Star-Shaped Differentiable Functions and Star-Shaped Differentials (Communications in Mathematical Research, 2021)
Based on the isomorphism between the space of star-shaped sets and thespace of continuous positively homogeneous real-valued functions, the star-shaped differential of a directionally differentiable function is defined. Formulas for star-shapeddifferential of a pointwise maximum and a pointwise minimum of a finite number ofdirectionally differentiable functions, and a composite of two directionally differentiable functions are derived. Furthermore, the mean-value theorem for a directionallydifferentiable function is demonstrated.
Keywords: The space of star-shaped sets; gauge function; isometrical isomorphism; directionally differentiable function; star-shaped differential; mean-value theorem.