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

Voir la notice de l'article provenant de la source Math-Net.Ru

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. 
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     title = {Completions of {Convex} {Families} of {Convex} {Bodies}},
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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/

[1] K. Kaveh, A. G. Khovanskii, “Newton–Okounkov bodies, semigroups of integral points, graded algebras and intersection theory”, Ann. of Math. (2) (to appear)

[2] V. V. Prasolov, V. M. Tikhomirov, Geometriya, MTsNMO, M., 2007