Completions of convex families of convex bodies. (English. Russian original) Zbl 1287.52010
Math. Notes 91, No. 3, 415-429 (2012); translation from Mat. Zametki 91, No. 3, 440-458 (2012).
Let \(\Delta: \mathbb{Z}^n_+\rightrightarrows \mathbb{R}^k_+\) be a set-valued mapping which maps every integer \(m\in\mathbb{Z}^n_+\) on a compact convex body \(\Delta_m\subset\mathbb{R}^k_+\) with the following properties:
1) \(\Delta_{m_1}+ \Delta_{m_2}\subseteq \Delta_{m_1+m_2}\) for \(m_1,m_2\in \mathbb{Z}^n_+\),
2) \(\Delta_{qm}= q\Delta_m\) for \(q\geq 0\) and \(m\in\mathbb{Z}^n_+\),
3) there exists a continuous function \(\phi: \mathbb{R}^n_+\to \mathbb{R}_+\) such that \(\phi(m)\) is the volume of \(\Delta_m\).
The question arises if the mapping \(\Delta\) can be extended continuously to the entire octant \(\mathbb{R}^n_+\). In the paper, the author gives a positive answer.
1) \(\Delta_{m_1}+ \Delta_{m_2}\subseteq \Delta_{m_1+m_2}\) for \(m_1,m_2\in \mathbb{Z}^n_+\),
2) \(\Delta_{qm}= q\Delta_m\) for \(q\geq 0\) and \(m\in\mathbb{Z}^n_+\),
3) there exists a continuous function \(\phi: \mathbb{R}^n_+\to \mathbb{R}_+\) such that \(\phi(m)\) is the volume of \(\Delta_m\).
The question arises if the mapping \(\Delta\) can be extended continuously to the entire octant \(\mathbb{R}^n_+\). In the paper, the author gives a positive answer.
Reviewer: Jörg Thierfelder (Ilmenau)
MSC:
| 52A41 | Convex functions and convex programs in convex geometry |
| 47H04 | Set-valued operators |
| 49J53 | Set-valued and variational analysis |
| 26E25 | Set-valued functions |
| 52A99 | General convexity |
References:
| [1] | K. Kaveh and A. G. Khovanskii, ”Newton-Okounkov bodies, semigroups of integral points, graded algebras and intersection theory,” submitted for publication in Ann. of Math. (2). · Zbl 1270.14022 |
| [2] | V. V. Prasolov and V. M. Tikhomirov, Geometry (MTsNMO, Moscow, 2007) [in Russian]. · Zbl 0977.51001 |
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