In mathematics, more precisely in the theory of functions of several complex variables, a pseudoconvex set is a special type of open set in the n-dimensional complex space Cn. Pseudoconvex sets are important, as they allow for classification of domains of holomorphy.


be a domain, that is, an open connected subset. One says that is pseudoconvex (or Hartogs pseudoconvex) if there exists a continuous plurisubharmonic function on such that the set

is a relatively compact subset of for all real numbers In other words, a domain is pseudoconvex if has a continuous plurisubharmonic exhaustion function. Every (geometrically) convex set is pseudoconvex. However, there are pseudoconvex domains which are not geometrically convex.

When has a (twice continuously differentiable) boundary, this notion is the same as Levi pseudoconvexity, which is easier to work with. More specifically, with a boundary, it can be shown that has a defining function, i.e., that there exists which is so that , and . Now, is pseudoconvex iff for every and in the complex tangent space at p, that is,

, we have

The definition above is analogous to definitions of convexity in Real Analysis.

If does not have a boundary, the following approximation result can be useful.

Proposition 1 If is pseudoconvex, then there exist bounded, strongly Levi pseudoconvex domains with (smooth) boundary which are relatively compact in , such that

This is because once we have a as in the definition we can actually find a C exhaustion function.

The case n = 1Edit

In one complex dimension, every open domain is pseudoconvex. The concept of pseudoconvexity is thus more useful in dimensions higher than 1.

See alsoEdit


  • Bremermann, H. J. (1956). "Complex Convexity". Transactions of the American Mathematical Society. 82 (1): 17–51. doi:10.1090/S0002-9947-1956-0079100-2. JSTOR 1992976.
  • Lars Hörmander, An Introduction to Complex Analysis in Several Variables, North-Holland, 1990. (ISBN 0-444-88446-7).
  • Steven G. Krantz. Function Theory of Several Complex Variables, AMS Chelsea Publishing, Providence, Rhode Island, 1992.
  • Siu, Yum-Tong (1978). "Pseudoconvexity and the problem of Levi". Bulletin of the American Mathematical Society. 84 (4): 481–513. doi:10.1090/S0002-9904-1978-14483-8. MR 0477104.
  • Catlin, David (1983). "Necessary Conditions for Subellipticity of the  -Neumann Problem". Annals of Mathematics. 117 (1): 147–171. doi:10.2307/2006974. JSTOR 2006974.

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