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Volume 26, Issue 1
Star-Shaped Differentiable Functions and Star-Shaped Differentials

Shaorong Pan, Hongwei Zhang & Liwei Zhang

Commun. Math. Res., 26 (2010), pp. 41-52.

Published online: 2021-05

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  • Abstract

Based on the isomorphism between the space of star-shaped sets and the space of continuous positively homogeneous real-valued functions, the star-shaped differential of a directionally differentiable function is defined. Formulas for star-shaped differential of a pointwise maximum and a pointwise minimum of a finite number of directionally differentiable functions, and a composite of two directionally differentiable functions are derived. Furthermore, the mean-value theorem for a directionally differentiable function is demonstrated.

  • Keywords

The space of star-shaped sets, gauge function, isometrical isomorphism, directionally differentiable function, star-shaped differential, mean-value theorem.

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COPYRIGHT: © Global Science Press

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@Article{CMR-26-41, author = {Pan , ShaorongZhang , Hongwei and Zhang , Liwei}, title = {Star-Shaped Differentiable Functions and Star-Shaped Differentials}, journal = {Communications in Mathematical Research }, year = {2021}, volume = {26}, number = {1}, pages = {41--52}, abstract = {

Based on the isomorphism between the space of star-shaped sets and the space of continuous positively homogeneous real-valued functions, the star-shaped differential of a directionally differentiable function is defined. Formulas for star-shaped differential of a pointwise maximum and a pointwise minimum of a finite number of directionally differentiable functions, and a composite of two directionally differentiable functions are derived. Furthermore, the mean-value theorem for a directionally differentiable function is demonstrated.

}, issn = {2707-8523}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/cmr/19176.html} }
TY - JOUR T1 - Star-Shaped Differentiable Functions and Star-Shaped Differentials AU - Pan , Shaorong AU - Zhang , Hongwei AU - Zhang , Liwei JO - Communications in Mathematical Research VL - 1 SP - 41 EP - 52 PY - 2021 DA - 2021/05 SN - 26 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/cmr/19176.html KW - The space of star-shaped sets, gauge function, isometrical isomorphism, directionally differentiable function, star-shaped differential, mean-value theorem. AB -

Based on the isomorphism between the space of star-shaped sets and the space of continuous positively homogeneous real-valued functions, the star-shaped differential of a directionally differentiable function is defined. Formulas for star-shaped differential of a pointwise maximum and a pointwise minimum of a finite number of directionally differentiable functions, and a composite of two directionally differentiable functions are derived. Furthermore, the mean-value theorem for a directionally differentiable function is demonstrated.

Pan , ShaorongZhang , Hongwei and Zhang , Liwei. (2021). Star-Shaped Differentiable Functions and Star-Shaped Differentials. Communications in Mathematical Research . 26 (1). 41-52. doi:
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