Quaternionic CR Geometry

Modelled on a real hypersurface in a quaternionic manifold, we introduce a quaternionic analogue of CR structure, called quaternionic CR structure. We define the strong pseudoconvexity of this structure as well as the notion of quaternionic pseudohermitian structure. Following the construction of the Tanaka-Webster connection in complex CR geometry, we construct a canonical connection associated with a quaternionic pseudohermitian structure, when the underlying quaternionic CR structure satisfies the ultra-pseudoconvexity which is stronger than the strong pseudoconvexity. Comparison to Biquard's quaternionic contact structure is also made.Comment: 41 page

Rigid characterizations of pseudoconvex domains

We prove that an open set $D$ in \C^n is pseudoconvex if and only if for any $z\in D$ the largest balanced domain centered at $z$ and contained in $D$ is pseudoconvex, and consider analogues of that characterization in the linearly convex case.Comment: v2: Proposition 14 is improved; v3: Example 15 and the proof of Proposition 14 are change

Regular versus singular order of contact on pseudoconvex hypersurfaces

The singular and regular type of a point on a real hypersurface $\mathcal H$ in $\mathbb C^n$ are shown to agree when the regular type is strictly less than 4. If $\mathcal H$ is pseudoconvex, we show they agree when the regular type is 4. A non-pseudoconvex example is given where the regular type is 4 and the singular type is infinite

Optimality conditions for scalar and vector optimization problems with quasiconvex inequality constraints

Let X be a real linear space, X0 X a convex set, Y and Z topological real linear spaces. The constrained optimization problem minCf(x), g(x) 2 -K is considered, where f : X0 ! Y and g : X0 ! Z are given (nonsmooth) functions, and C Y and K Z are closed convex cones. The weakly efficient solutions (w-minimizers) of this problem are investigated. When g obeys quasiconvex properties, first-order necessary and first-order sufficient optimality conditions in terms of Dini directional derivatives are obtained. In the special case of problems with pseudoconvex data it is shown that these conditions characterize the global w-minimizers and generalize known results from convex vector programming. The obtained results are applied to the special case of problems with finite dimensional image spaces and ordering cones the positive orthants, in particular to scalar problems with quasiconvex constraints. It is shown, that the quasiconvexity of the constraints allows to formulate the optimality conditions using the more simple single valued Dini derivatives instead of the set valued ones. Key words: Vector optimization, nonsmooth optimization, quasiconvex vector functions, pseudoconvex vector functions, Dini derivatives, quasiconvex programming, Kuhn-Tucker conditions..