## Arkiv för Matematik

• Ark. Mat.
• Volume 53, Number 2 (2015), 203-236.

### Extremal functions for real convex bodies

#### Abstract

We study the smoothness of the Siciak–Zaharjuta extremal function associated to a convex body in $\mathbb{R}^{2}$. We also prove a formula relating the complex equilibrium measure of a convex body in $\mathbb{R}^{n}$ (n≥2) to that of its Robin indicatrix. The main tool we use is extremal ellipses.

#### Note

The third author was partially supported by University of Auckland grant 3704154.

#### Article information

Source
Ark. Mat., Volume 53, Number 2 (2015), 203-236.

Dates
First available in Project Euclid: 30 January 2017

https://projecteuclid.org/euclid.afm/1485802714

Digital Object Identifier
doi:10.1007/s11512-014-0207-6

Mathematical Reviews number (MathSciNet)
MR3391168

Zentralblatt MATH identifier
1327.32045

Rights

#### Citation

Burns, Daniel M.; Levenberg, Norman; Ma‘u, Sione. Extremal functions for real convex bodies. Ark. Mat. 53 (2015), no. 2, 203--236. doi:10.1007/s11512-014-0207-6. https://projecteuclid.org/euclid.afm/1485802714

#### References

• Baran, M., Plurisubharmonic extremal functions and complex foliations for the complement of convex sets in $\mathbb{R}^{n}$, Michigan Math. J. 39 (1992), 395–404.
• Bedford, E. and Ma‘u, S., Complex Monge–Ampère of a maximum, Proc. Amer. Math. Soc. 136 (2008), 95–101.
• Bedford, E. and Taylor, B. A., The Dirichlet problem for a complex Monge–Ampère equation, Invent. Math. 37 (1976), 1–44.
• Bedford, E. and Taylor, B. A., A new capacity for plurisubharmonic functions, Acta Math. 149 (1982), 1–40.
• Bedford, E. and Taylor, B. A., Uniqueness for the complex Monge–Ampère equation for functions of logarithmic growth, Indiana Univ. Math. J. 38 (1989), 455–469.
• Bloom, T., Levenberg, N. and Ma‘u, S., Robin functions and extremal functions, Ann. Polon. Math. 80 (2003), 55–84.
• Burns, D., Levenberg, N. and Ma‘u, S., Pluripotential theory for convex bodies in $\mathbb{R}^{n}$, Math. Z. 250 (2005), 91–111.
• Burns, D., Levenberg, N. and Ma‘u, S., Exterior Monge–Ampère solutions, Adv. Math. 222 (2009), 331–358.
• Burns, D., Levenberg, N., Ma‘u, S. and Revesz, Sz., Monge–Ampère measures for convex bodies and Bernstein–Markov type inequalities, Trans. Amer. Math. Soc. 362 (2010), 6325–6340.
• Guan, B., On the regularity of the pluricomplex Green functions, Int. Math. Res. Not. 22 (2007), 19 pp.
• Klimek, M., Pluripotential Theory, Oxford University Press, New York, 1991.
• Lempert, L., La métrique de Kobayashi et la représentation des domaines sur la boule, Bull. Soc. Math. France 109 (1981), 427–474.
• Lempert, L., Intrinsic distances and holomorphic retracts, in Complex Analysis and Applications81 (Varna, 1981), pp. 341–364, Publ. House Bulgar. Acad. Sci., Sofia, 1984.
• Lempert, L., Symmetries and other transformations of the complex Monge–Ampère equation, Duke Math. J. 52 (1985), 869–885.
• Lempert, L., Holomorphic invariants, normal forms, and the moduli space of convex domains, Ann. of Math. 128 (1988), 43–78.
• Lundin, M., The extremal PSH for the complement of convex, symmetric subsets of $\mathbb{R}^{n}$, Michigan Math. J. 32 (1985), 197–201.
• Momm, S., An extremal plurisubharmonic function associated to a convex pluricomplex Green function with pole at infinity, J. Reine Angew. Math. 471 (1996), 139–163.
• Ransford, T., Potential Theory in the Complex Plane, Cambridge University Press, Cambridge, 1995.