It is proved that the simplex is a strict local minimum for the volume product, 𝒫(K)=min z∈int(K)|K||Kz|, in the Banach–Mazur space of n-dimensional (classes of) convex bodies. Linear local stability in the neighborhood of the simplex is proved as well. The proof consists of an extension to the non-symmetric setting of methods that were recently introduced by Nazarov, Petrov, Ryabogin and Zvavitch, as well as proving results of independent interest concerning stability of square order of volumes of polars of non-symmetric convex bodies.

Let f(x) be a monic polynomial of degree n with complex coefficients, which factors as f(x) = g(x)h(x), where g and h are monic. Let be the maximum of on the unit circle. We prove that , where β = M(P0) = 1 38135 …, where P0 is the polynomial P0(x, y) = 1 + x + y and δ = M(P1) = 1 79162…, where P1(x, y) = 1 + x + y - xy, and M denotes Mahler's measure. Both inequalities are asymptotically sharp as n → ∞.

We give improved bounds for our theorem in [W. T. Gowers and J. Wolf, The true complexity of a system of linear equations. Proc. London Math. Soc. (3) 100 (2010), 155–176], which shows that a system of linear forms on 𝔽np with squares that are linearly independent has the expected number of solutions in any linearly uniform subset of 𝔽np. While in [W. T. Gowers and J. Wolf, The true complexity of a system of linear equations. Proc. London Math. Soc. (3) 100 (2010), 155–176] the dependence between the uniformity of the set and the resulting error in the average over the linear system was of tower type, we now obtain a doubly exponential relation between the two parameters. Instead of the structure theorem for bounded functions due to Green and Tao [An inverse theorem for the Gowers U3(G) norm. Proc. Edinb. Math. Soc. (2) 51 (2008), 73–153], we use the Hahn–Banach theorem to decompose the function into a quadratically structured plus a quadratically uniform part. This new decomposition makes more efficient use of the U3 inverse theorem [B. J. Green and T. Tao, An inverse theorem for the Gowers U3(G) norm. Proc. Edinb. Math. Soc. (2) 51 (2008), 73–153].

With each non-empty compact convex subset K of Ed is associated a Steiner point, s(K), defined by

where u is a variable unit vector, a is a fixed unit vector, H(u, K) is the supporting function of K and dw is an element of surface area of the unit sphere Sd-1 centred at the origin (see [2]). For notational convenience, we put s(Ø) = 0.

In an unpublished, dissertation Cleaver [1] proved the following

Theorem 1. If L is a lattice in euclidean four-space R4 of determinant d(L) = 1 and with no pair of its points within unit distance apart then any four-sphere of radius 1 contains a point of L.

We show that for every smooth curve in Rn, there is a quadrilateral with equal sides and equal diagonals whose vertices lie on the curve. In the case of a smooth plane curve, this implies that the curve admits an inscribed square, strengthening a theorem of Schnirelmann and Guggenheimer. “Smooth” means having a continuously turning tangent. We give a weaker condition which is still sufficient for the existence of an inscribed square in a plane curve, and which is satisfied if the curve is convex, if it is a polygon, or (with certain restrictions) if it is piecewise of class C1. For other curves, the question remains open.

We prove a normalized version of the restricted invertibility principle obtained by Spielman and Srivastava in [An elementary proof of the restricted invertibility theorem. Israel J. Math. 190 (2012), 83–91]. Applying this result, we get a new proof of the proportional Dvoretzky–Rogers factorization theorem recovering the best current estimate in the symmetric setting while we improve the best known result in the non-symmetric case. As a consequence, we slightly improve the estimate for the Banach–Mazur distance to the cube: the distance of every $n$-dimensional normed space from ${ \ell }_{\infty }^{n} $ is at most $\mathop{(2n)}\nolimits ^{5/ 6} $. Finally, using tools from the work of Batson et al in [Twice-Ramanujan sparsifiers. In STOC’09 – Proceedings of the 2009 ACM International Symposium on Theory of Computing, ACM (New York, 2009), 255–262], we give a new proof for a theorem of Kashin and Tzafriri [Some remarks on the restriction of operators to coordinate subspaces. Preprint, 1993] on the norm of restricted matrices.

In this note we shall be mainly concerned with convex bodies of constant width in En that are invariant under the group of congruences that leave invariant a regular simplex with its centre of gravity at the origin. We first show that there are many such convex bodies. This follows, by showing that any set S ot diameter 1 that is invariant under a group of congruences about the origin, is contained in a convex body of constant width 1 that is invariant under the group.

Introduction. Polyhedra in 3-dimensional hyperbolic space which give rise to discrete groups generated by reflections in their faces have been investigated in [14], [17], [29] and in the case of tetrahedra there are precisely nine compact non-congruent ones with dihedral angles integral submultiples of π [14]. These polyhedral groups give rise to hyperbolic 3-orbifolds and examples of these have been studied, for example, in [3], [15], [18], [24], [25].

A famous conjecture of Hardy and Littlewood [4] stated that all sufficiently large integers n could be represented in the form

where p is a rational prime and x, y are integers. G. K. Stanley [9] showed that this result held for “almost all” integers n if one assumed a hypothesis concerning the zeros of L-functions similar to, though weaker than, the extended Riemann hypothesis.

Existence criteria are presented for non-linear boundary value problems on the half line. In particular, the theory includes a problem in the theory of colloids and a problem arising in the unsteady flow of a gas through a semi-infinite porous medium.

A graph consists of a set of vertices some pairs of which are joined by a single edge. A tree is a graph with the property that each pair of vertices is connected by precisely one path, i.e., a sequence of distinct vertices joined consecutively by edges. The complexity c of a graph G(n, k) with n vertices and k edges is the number of trees with n vertices which are subgraphs of G(n, k). The distribution of c over the class of all graphs G(n, k) is of physical interest because it throws light on the classical many-body problem. (See, e.g. [9].) Ford and Uhlenbeck [3] gave numerical data which suggested that the distribution of c tends to normality for increasing n if k is near No moments higher than the first were known in general and they remarked in [4] that even “the second would be worth knowing”. The main object in this paper is to derive a formula for the second moment of c.

We study the system of Maxwell equations for a periodic composite dielectric medium with components whose dielectric permittivities ${\it\epsilon}$ have a high degree of contrast between each other. We assume that the ratio between the permittivities of the components with low and high values of ${\it\epsilon}$ is of the order ${\it\eta}^{2}$, where ${\it\eta}>0$ is the period of the medium. We determine the asymptotic behaviour of the electromagnetic response of such a medium in the “homogenization limit”, as ${\it\eta}\rightarrow 0$, and derive the limit system of Maxwell equations in $\mathbb{R}^{3}$. Our results extend a number of conclusions of a paper by Zhikov [On gaps in the spectrum of some divergent elliptic operators with periodic coefficients. St. Petersburg Math. J.16(5) (2004), 719–773] to the case of the full system of Maxwell equations.

Erdös and Straus have conjectured that for every integer n > 1,

is soluble in positive integers x, y, z. Schinzel has conjectured that for every a > 0 if n > no(a),

is soluble in positive integers x, y, z.

A study is made of the length L(h, k) of the Euclidean algorithm for determining the g.c.d. of two polynomials h, k in [X], a finite field. We obtain exact formulae for the number of pairs with a fixed length N which lie in a given range, as well as the average length and variance of the Euclidean algorithm for such pairs.

For a fixed integer k ≥ 2 suppose we have functions Li (x) satisfying the following conditions.

A convex polytope is a zonotope, if, and only if, its support function satisfies Hlawka's inequality. It follows that a finite dimensional real space with piecewise linear norm is isometrically isomorphic to a subspace of an L1 space, if, and only if, it has the quadrilateral property

In this note we prove:

Theorem. Let Λ be a normal number field of absolute degree n. Then there exist infinitely many normal number fields K with the following properties.

(i) The Galois group of K over the rational field Q is the symmetric group Sn on n symbols.

In this paper we shall set out the generalization, for n-dimensional space Sn, of some recent results about complete quadrics and complete collineations in S2, S3 and S4. For the results about complete conies in S2, originally introduced by Study [1], we refer the reader to papers by Severi ([2], [3]), van der Waerden [4], Semple [5]; for those about complete quadrics in S3, to Semple ([6], [7]); for the extension to S4 to Alguneid [8]; for the general concept of complete collineations in Sn, and for results in S2 and S3, to Semple [9].

All of our work takes place in Ed, d-dimensional Euclidean space, with unit ball B. Unless specifically noted to the contrary, all sets will be presumed to be closed and convex. If X is a subset of a sphere S with centre p, we will say X is spherically convex if X is contained in some open hemisphere of S and if the cone generated by X with vertex p is convex. The distance between two points x, y ε Ed will be denoted |x − y|. If K1K2 are two convex sets, ρ(K1, K2) will mean the usual Hausdorff distance between them.