Geodesic
Completions of Convex Families of Convex Bodies
Matematičeskie zametki, Tome 91 (2012) no. 3, pp. 440-458

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The paper discusses the existence of a continuous extension of functions that are defined on subsets of $\mathbb R^n$ and whose values are convex bodies in $\mathbb R^n$. This problem arose in convex geometry in connection with the notion, recently introduced in algebraic geometry, of convex Newton–Okounkov bodies.
Keywords: Newton–Okounkov body, convex-valued function, continuity, extension. 
@article{MZM_2012_91_3_a10,
     author = {A. G. Khovanskii},
     title = {Completions of {Convex} {Families} of {Convex} {Bodies}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {440--458},
     publisher = {mathdoc},
     volume = {91},
     number = {3},
     year = {2012},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2012_91_3_a10/}
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A. G. Khovanskii. Completions of Convex Families of Convex Bodies. Matematičeskie zametki, Tome 91 (2012) no. 3, pp. 440-458. http://geodesic.mathdoc.fr/item/MZM_2012_91_3_a10/