Volume 12, Issue 4

We investigate barycenters of probability measures on proper Alexandrov spaces of curvature bounded below, and show that they enjoy several properties relevant to or different from those in metric spaces of curvature bounded above. We prove the reverse variance inequality, and show that the push forward of a measure to the tangent cone at its barycenter has the flat support.

We consider the isoperimetric problem in planar sectors with density r p , and with density a > 1 inside the unit disk and 1 outside. We characterize the solutions as a function of the sector angle. We also solve the isoperimetric problem in ℝ n with density r p , p < 0

Two dependent examples are presented: 1. Two convex polytopes in ℝ 3 such that for each pair of their parallel facets, one of the facets fits strictly into the other. (The example gives a refinement of A. D. Alexandrov’s uniqueness theorem for convex polytopes.) 2. A pointed tiling of the two-sphere S 2 generated by a Laman-plus-one graph which can be regularly triangulated without adding extra vertices. The construction uses the combinatorial rigidity theory of spherically embedded graphs and the relationship between the theory of pseudo triangulations and the theory of hyperbolic virtual polytopes.

Given a warped product space ℝ ×f N with logarithmically convex warping function f, we prove a relative isoperimetric inequality for regions bounded between a subset of a vertical fiber and its image under an almost everywhere differentiable mapping in the horizontal direction. In particular, given a k-dimensional region F ⊂ {b} × N , and the horizontal graph C ⊂ ℝ ×f N of an almost everywhere differentiable map over F, we prove that the k-volume of C is always at least the k-volume of the smooth constant height graph over F that traps the same (1 + k)-volume above F as C. We use this to solve a Dido problem for graphs over vertical fibers, and show that, if the warping function is unbounded on the set of horizontal values above a vertical fiber, the volume trapped above that fiber by a graph C is no greater than the k-volume of C times a constant that depends only on the warping function.

We prove that the UPC (uniformly polynomially cuspidal) condition holds in an ominimal structure generated by some convergent generalized power series. The result is obtained by geometric and analytic methods. We get automatically some consequences. First of all, we obtain a new large class of sets with cusps satisfying Markov’s inequality (the class is strictly larger than the class of compact, fat and globally subanalytic sets). It may be useful from the point of view of approximation theory (multivariate polynomial inequalities) and constructive function theory. Our main result finds also practical application in complex analysis - it gives new examples of sets with the HCP property (Hölder continuity property of the Siciak extremal function).

We prove uniqueness results for conics of minimal area that enclose a compact, fulldimensional subset of the elliptic plane. The minimal enclosing conic is unique if its center or axes are prescribed. Moreover, we provide sufficient conditions on the enclosed set that guarantee uniqueness without restrictions on the enclosing conics. Similar results are formulated for minimal enclosing conics of line sets as well.

In this paper we study a class of unbounded domains in ℂ 2 which are invariant with respect to translations in some fixed real direction. Such domains will be called semi-tubes. We prove various properties of such domains, e.g., dependence of pseudoconvexity on the position of the base of the domain in ℝ 3 ≃ ℂ × ℝ the Hartogs type phenomenon on holomorphic extension of CR functions, and give concrete examples of semi-tubes over tori in ℝ 3 ≃ℂ × ℝ and see how the position of the tori influences the pseudoconvexity of the semi-tube.

This paper contributes to the classification of flag transitive c:F 4 -geometries. Here, under some mild extra conditions, we determine those geometries in which the point-residue is a building of type F 4 (2) and we show that the automorphism groups are 2 E 6 (2) : 2 or 3. 2 E 6 (2) : 2 or E 6 (2) : 2 or 2 26 : F 4 (2). This work was started by Corinna Wiedorn. In this paper we follow her ideas and just show that the corresponding amalgams are uniquely determined.

We classify the Majorana representations of the simple group L 3 (2). It turns out that, subject to the so-called (2A)-condition, this group possesses precisely two Majorana representations. The dimensions of these representations are 21 and 49, and both representations are based on the embeddings of L 3 (2) into the Monster as 2A-generated subgroups.

In [10], it was shown that small t-fold (n - k)-blocking sets in PG(n; q), q = p h , p prime, h ≥ 1, intersect every k-dimensional space in t (mod p) points. We characterize in this article all t-fold (n - k)-blocking sets in PG(n,q), q square, q ≥ 661, t<c p q 1/6 /2 |B|<tq n-k +2tq n-k-1 √q intersecting every k-dimensional space in t (mod √q ) points.