There are only 10 Euclidean forms, that is flat closed three dimensional manifolds: six are orientable and four are non-orientable. The aim of this paper is to describe all types of n-fold coverings over the orientable Euclidean manifolds G(3) and G(5), and calculate the numbers of non-equivalent coverings of each type. The manifolds G(3) and G(5) are uniquely determined among other forms by their homology groups H-1 (G(3)) = Z(3) x Z and H-1 (G(5)) = Z.We classify subgroups in the fundamental groups pi(1) (G(3) ) and pi(1) (G(5)) up to isomorphism. Given index n, we calculate the numbers of subgroups and the numbers of conjugacy classes of subgroups for each isomorphism type and provide the Dirichlet generating functions for the above sequences. (C) 2020 Elsevier Inc. All rights reserved.

A union of an arrangement of affine hyperplanes H in R-d is the real algebraic variety associated to the principal ideal generated by the polynomial pH given as the product of the degree one polynomials which define the hyperplanes of the arrangement. A finite Borel measure on R-d is bisected by the arrangement of affine hyperplanes H if the measure on the "non-negative side" of the arrangement {x is an element of R-d : pH(x) >= 0} is the same as the measure on the "non-positive" side of the arrangement {x is an element of R-d : pH(x) <= 0}. In 2017 Barba, Pilz & Schnider considered special, as well as modified cases of the following measure partition hypothesis: For a given collection of j finite Borel measures on R-d there exists a k-element affine hyperplane arrangement that bisects each of the measures into equal halves simultaneously. They showed that there are simultaneous bisections in the case when d = k = 2 and j = 4. Furthermore, they conjectured that every collection of j measures on R-d can be simultaneously bisected with a k-element affine hyperplane arrangement provided that d >= inverted right perpendicular j/k inverted left perpendicular. The conjecture was confirmed in the case when d >= j/k = 2(a) by Hubard and Karasev in 2018. In this paper we give a different proof of the Hubard and Karasev result using the framework of Blagojevi ' c, Frick, Haase & Ziegler (2016), based on the equivariant relative obstruction theory of tom Dieck, which was developed for handling the Grunbaum-Hadwiger-Ramos hyperplane measure partition problem. Furthermore, this approach allowed us to prove even more, that for every collection of 2(a)(2h + 1)+l measures on R2a+l, where 1 <= l <= 2(a) - 1, there exists a (2h + 1)-element affine hyperplane arrangement that bisects all of them simultaneously. Our result was extended to the case of spherical arrangements and reproved by alternative methods in a beautiful way by Crabb [8].

We break the exponential barrier for triangulations of the real projective space, constructing a trianglation of RPn with e((1/2 + o(1))root nlogn) vertices.

We study the 'no-dimension' analogue of Caratheodory's theorem in Banach spaces. We prove such a result together with its colorful version for uniformly smooth Banach spaces. It follows that uniform smoothness leads to a greedy de-randomization of Maurey's classical lemma Pisier (in: Seminaire Analyse fonctionnelle (dit "Maurey-Schwartz"), 1980), which is itself a 'no-dimension' analogue of Caratheodory's theorem with a probabilistic proof. We find the asymptotically tight upper bound on the deviation of the convex hull from the k-convex hull of a bounded set in L-p with 1 < p <= 2 and get asymptotically the same bound as in Maurey's lemma for L-p with 1 < p < infinity.

We use symplectic techniques to obtain partial results on Mahler's conjecture about the product of the volume of a convex body and the volume of its polar. We confirm the conjecture for hyperplane sections or projections of l(p)-balls or the Hanner polytopes.

We study properties of the volume of projections of the n-dimensional cross-polytope lozenge(n) = {x is an element of R-n vertical bar vertical bar x(1)vertical bar +...+ vertical bar x(n)vertical bar <= 1}. We prove that the projection of lozenge(n) onto a k-dimensional coordinate subspace has the maximum possible volume for k = 2 and for k = 3. We obtain the exact lower bound on the volume of such a projection onto a two-dimensional plane. Also, we show that there exist local maxima which are not global ones for the volume of a projection of lozenge(n) onto a k-dimensional subspace for any n > k >= 2. (C) 2021 Elsevier B.V. All rights reserved.

We show that provided that n is a power of two, any nD measures in R-n can be bisected by an arrangement of D hyperplanes.

Rudelson's theorem states that if for a set of unit vectors u(i) and positive weights c(i), we have that Sigma c(i)u(i) circle times u(i) is the identity operator I on R-d, then the sum of a random sample of Cd In d of these diadic products is close to I. The In d term cannot be removed.On the other hand, the recent fundamental result of Batson, Spielman and Srivastava and its improvement by Marcus, Spielman and Srivastava show that the In d term can be removed, if one wants to show the existence of a good approximation of I as the average of a few diadic products. It is known that essentially the same proof as Rudelson's yields a more general statement about the average of positive semidefinite matrices.First, we give an example of an average of positive semi-definite matrices where there is no approximation of this average by Cd elements. Thus, the result of Batson, Spielman and Srivastava cannot be extended to this wider class of matrices.Next, we present a stability version of Rudelson's result on positive semi-definite matrices, and thus, extend it to certain non-symmetric matrices. This yields applications to the study of the Banach-Mazur distance of convex bodies. Finally, we show that in some cases, one needs to take a subset of the vectors of order d(2) to approximate the identity. (C) 2020 The Author(s). Published by Elsevier Inc.

There are only 10 Euclidean forms, that is flat closed three-dimensional manifolds: six are orientable and four are non-orientable. The aim of this paper is to describe all types of n-fold coverings over orientable Euclidean manifolds G(2) and G(4) and calculate the numbers of nonequivalent coverings of each type. We classify subgroups in the fundamental groups pi(1)(G(2)) and pi(1)(G(4)) up to isomorphism and calculate the numbers of conjugated classes of each type of subgroups for index n. The manifolds and are uniquely determined among the others orientable forms by their homology groups H-1(G(2)) = Z(2) x Z(2) x Z and H-1(G4) = Z(2) x Z.

We apply the billiard technique to deduce some results on Viterbo's conjectured inequality between the volume of a convex body and its symplectic capacity. We show that the product of a permutohedron and a simplex (properly related to each other) delivers equality in Viterbo's conjecture. Using this result as well as previously known equality cases, we prove some special cases of Viterbo's conjecture and interpret them as isoperimetric-like inequalities for billiard trajectories.