A homogeneous real polynomial $p$ is {\em hyperbolic} with respect to a given vector $d$ if the univariate polynomial $t \mapsto p(x-td)$ has all real roots for all vectors $x$. Motivated by partial differential equations, G{\aa}rding proved in 1951 that the largest such root is a convex function of $x$, and showed various ways of constructing new hyperbolic polynomials. We present a powerful new such construction, and use it to generalize G{\aa}rding's result to arbitrary symmetric functions of the roots. Many classical and recent inequalities follow easily. We develop various convex-analytic tools for such symmetric functions, of interest in interior-point methods for optimization problems over related cones.

Lattices and $\ZZ$-modules in Euclidean space possess an infinitude of subsets that are images of the original set under similarity transformation. We classify such self-similar images according to their indices for certain 4D examples that are related to 4D root systems, both crystallographic and non-crystallographic. We encapsulate their statistics in terms of Dirichlet series generating functions and derive some of their asymptotic properties.

Partial answers are given to two questions. When does a lattice $\Lambda$ contain a sublattice $\Lambda'$ of index $N$ that is geometrically similar to $\Lambda$? When is the sublattice ``clean'', in the sense that the boundaries of the Voronoi cells for $\Lambda'$ do not intersect $\Lambda$?

In this article, we derive a Brunn-Minkowski type theorem for sets bearing some relation to the causal structure on the Minkowski spacetime $\mathbb{L}^{n+1}$. We also present an isoperimetric inequality in the Minkowski spacetime $\mathbb{L}^{n+1}$ as a consequence of this Brunn-Minkowski type theorem.

Let $M$ be a convex body such that the boundary has positive curvature. Then by a well developed theory dating back to Landau and Hlawka for large $\lambda$ the number of lattice points in $\lambda M$ is given by $G(\lambda M) =V(\lambda M) + O(\lambda^{d-1-\varepsilon (d)})$ for some positive $\varepsilon(d)$. Here we give for general convex bodies the weaker estimate \[ \left| G(\lambda M) -V(\lambda M) \right | \le \frac{1}{2} S_{\Z^d}(M) \lambda^{d-1}+o(\lambda^{d-1}) \] where $S_{\Z^d}(M)$ denotes the lattice surface area of $M$. The term $S_{\Z^d}(M)$ is optimal for all convex bodies and $o(\lambda^{d-1})$ cannot be improved in general. We prove that the same estimate even holds if we allow small deformations of $M$. Further we deal with families $\{P_\lambda\}$ of convex bodies where the only condition is that the inradius tends to infinity. Here we have \[ \left| G(P_\lambda)-V(P_\lambda) \right| \le dV(P_\lambda,K;1)+o \bigl( S(P_\lambda) \bigr) \] where the convex body $K$ satisfies some simple condition, $V(P_\lambda,K;1)$ is some mixed volume and $S(P_\lambda)$ is the surface area of $P_\lambda$.

No abstract.

Let $K$ be a convex body in $\ed$ and denote by $\cn$ the set of centroids of $n$ non-overlapping translates of $K$. For $\varrho>0$, assume that the parallel body $\cocn+\varrho K$ of $\cocn$ has minimal volume. The notion of parametric density (see~\cite{Wil93}) provides a bridge between finite and infinite packings (see~\cite{BHW94} or~\cite{Hen}). It is known that there exists a maximal $\varrho_s(K)\geq 1/(32d^2)$ such that $\cocn$ is a segment for $\varrho<\varrho_s$ (see~\cite{BHW95}). We prove the existence of a minimal $\varrho_c(K)\leq d+1$ such that if $\varrho>\varrho_c$ and $n$ is large then the shape of $\cocn$ can not be too far from the shape of $K$. For $d=2$, we verify that $\varrho_s=\varrho_c$. For $d\geq 3$, we present the first example of a convex body with known $\varrho_s$ and $\varrho_c$; namely, we have $\varrho_s=\varrho_c=1$ for the parallelotope.

In this paper we introduce the concepts of hyperbolic and elliptic areas and prove uncountably many new geometric isoperimetric inequalities on the surfaces of constant curvature.