Abstract The purpose of this note is to give counterexamples defined over the real numbers to a question relevant in commutative algebra, concerning a containment relation between algebraic and symbolic powers of a homogeneous ideal.

Abstract Constructing constant width sets or, more generally, complete sets in Banach spaces seems to be a not so easy task. A new construction working in separable Banach spaces is presented, and Bavaud’s and Lachand-Robert and Oudet’s constructions of complete sets are extended to a more general context.

Abstract We determine the Euclidean distortion of finite generalized polygons by means of semidefinite programing.

Abstract We prove that among the quasi-simple Lie groups only the group SL4(ℝ) occurs as the multiplication group of 3-dimensional connected topological loops L. These loops L are homeomorphic to the sphere S3. Moreover, there does not exist any connected topological loop having an at most 8-dimensional quasi-simple Lie groups as its multiplication group.

Abstract We study the geometry of semitube domains in ℂ2, in particular we extend the result of Burgués and Dwilewicz for semitube domains by dropping the smoothness assumption. We also prove various properties of non-smooth pseudoconvex semitube domains, obtaining a relation between pseudoconvexity of a semitube domain and the number of components of its vertical slices. Finally, we present an example of a non-convex domain in ℂn such that its image under arbitrary isometries is pseudoconvex.

Abstract We study the syzygetic structure of projections of del Pezzo surfaces in order to construct singular Calabi-Yau threefolds. By smoothing those threefolds,we obtain new examples of Calabi-Yau threefolds with Picard group of rank 1. We also give an example of type II primitive contraction whose exceptional divisor is the blow-up of the projective plane at a point.

Abstract Ehrhart’s conjecture proposes a sharp upper bound on the volume of a convex body whose barycenter is its only interior lattice point. Recently, Berman and Berndtsson proved this conjecture for a class of rational polytopes including reflexive polytopes. In particular, they showed that the complex projective space has the maximal anticanonical degree among all toric Kähler- Einstein Fano manifolds. In this note, we prove that projective space is the only such toric manifold with maximal degree by proving the corresponding convex-geometric statement. We also discuss a generalized version of Ehrhart’s conjecture involving an invariant corresponding to the so-called greatest lower bound on the Ricci curvature.

Abstract We deal with different properties of a smooth and strictly convex body that depend on the behavior of the planar sections of the body parallel to and close to a given tangent plane. The first topic is boundary points where any given convex domain in the tangent plane can be approximated by a sequence of suitably rescaled planar sections (so-called p-universal points). In the second topic, the given convex body is the unit ball of a Minkowski space, and oscillation properties of bisectors and trisectors in that space are considered. In either case it turns out that geometrically irregular behavior is typical, when considered from the viewpoint of Baire category.

Abstract In the classical theory of several complex variables, holomorphic mappings are just n-tuples of holomorphic functions in m variables, with arbitrary n andm, and no relations between these functions are assumed. Some 30 years ago John Ryan introduced complex, or complexified, Clifford analysis which is, in a sense, the study of certain classes of holomorphic mappings where the components are not independent, and instead obey the relations generated by the Cauchy- Riemann and Dirac-type operators. In this paper, we take a closer look at this theory emphasizing some additional properties that holomorphic mappings satisfy in this context. Our attention is mostly restricted to the case of low dimensions where it is possible to identify new and interesting properties and to single out the special role played by bicomplex analysis.