We prove several sharp distortion and monotonicity theorems for spherically convex functions defined on the unit disk involving geometric quantities such as spherical length, spherical area, and total spherical curvature. These results can be viewed as geometric variants of the classical Schwarz lemma for spherically convex functions.

For any $n>1$ we determine the uniform and nonuniform lattices of the smallest covolume in the Lie group $\operatorname {\mathrm {Sp}}(n,1)$ . We explicitly describe them in terms of the ring of Hurwitz integers in the nonuniform case with n even, respectively, of the icosian ring in the uniform case for all $n>1$ .

Let $\Omega $ be a connected open set in the plane and $\gamma : [0,1] \to \overline {\Omega }$ a path such that $\gamma ((0,1)) \subset \Omega $ . We show that the path $\gamma $ can be “pulled tight” to a unique shortest path which is homotopic to $\gamma $ , via a homotopy h with endpoints fixed whose intermediate paths $h_t$ , for $t \in [0,1)$ , satisfy $h_t((0,1)) \subset \Omega $ . We prove this result even in the case when there is no path of finite Euclidean length homotopic to $\gamma $ under such a homotopy. For this purpose, we offer three other natural, equivalent notions of a “shortest” path. This work generalizes previous results for simply connected domains with simple closed curve boundaries.

Higher dimensional analogues of the modular group $\mathit{PSL}(2,\mathbb{Z})$ are closely related to hyperbolic reflection groups and Coxeter polyhedra with big symmetry groups. In this context, we develop a theory and dissection properties of ideal hyperbolic $k$-rectified regular polyhedra, which is of independent interest. As an application, we can identify the covolumes of the quaternionic modular groups with certain explicit rational multiples of the Riemann zeta value $\unicode[STIX]{x1D701}(3)$.

A rational polyhedron$P\subseteq {\mathbb{R^n}}$ is a finite union of simplexes in ${\mathbb{R^n}}$ with rational vertices. P is said to be $\mathbb Z$-homeomorphic to the rational polyhedron $Q\subseteq {\mathbb{R^{\it m}}}$ if there is a piecewise linear homeomorphism η of P onto Q such that each linear piece of η and η−1 has integer coefficients. When n=m, $\mathbb Z$-homeomorphism amounts to continuous $\mathcal{G}_n$-equidissectability, where $\mathcal{G}_n=GL(n,\mathbb Z) \ltimes \mathbb Z^{n}$ is the affine group over the integers, i.e., the group of all affinities on $\mathbb{R^{n}}$ that leave the lattice $\mathbb Z^{n}$ invariant. $\mathcal{G}_n$ yields a geometry on the set of rational polyhedra. For each d=0,1,2,. . ., we define a rational measure λd on the set of rational polyhedra, and show that any two $\mathbb Z$-homeomorphic rational polyhedra $$P\subseteq {\mathbb{R^n}}$$ and $Q\subseteq {\mathbb{R^{\it m}}}$ satisfy $\lambda_d(P)=\lambda_d(Q)$. $\lambda_n(P)$ coincides with the n-dimensional Lebesgue measure of P. If 0 ≤ dim P=d < n then λd(P)>0. For rational d-simplexes T lying in the same d-dimensional affine subspace of ${\mathbb{R^{\it n}}, $\lambda_d(T)$$ is proportional to the d-dimensional Hausdorff measure of T. We characterize λd among all unimodular invariant valuations.

We define an infinite class of fractals, called horizontally and vertically blocked labyrinth fractals, which are dendrites and special Sierpiński carpets. Between any two points in the fractal there is a unique arc α; the length of α is infinite and the set of points where no tangent to α exists is dense in α.

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.

In the first part we construct a subset H of positive measure in the unit interval and a zero-sequence {an} so that H contains no homothetic copy of {an}. In Theorem 2 we prove that if ε > 0 and a zero-sequence {an} are given then there exists a set A of measure less than ε so that covers the interval. An application of this result is Theorem 3: for any sequence {an} and ε > 0 there is a set H of measure 1 - ε such that for no N and c is {an + c}n ≥ N contained by H.