Geodesic
Criteria for Convexity of Closed Sets in Banach Spaces
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Optimal control and differential equations, Tome 304 (2019), pp. 205-220

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Criteria for the convexity of closed sets in general Banach spaces in terms of the Clarke and Bouligand tangent cones are proved. In the case of uniformly convex spaces, these convexity criteria are stated in terms of proximal normal cones. These criteria are used to derive sufficient conditions for the convexity of the images of convex sets under nonlinear mappings and multifunctions. 
@article{TM_2019_304_a12,
     author = {Yu. S. Ledyaev},
     title = {Criteria for {Convexity} of {Closed} {Sets} in {Banach} {Spaces}},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {205--220},
     publisher = {mathdoc},
     volume = {304},
     year = {2019},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2019_304_a12/}
}
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Yu. S. Ledyaev. Criteria for Convexity of Closed Sets in Banach Spaces. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Optimal control and differential equations, Tome 304 (2019), pp. 205-220. http://geodesic.mathdoc.fr/item/TM_2019_304_a12/