Related papers: The geodesic complexity of n-dimensional Klein bot…

A fundamental question for simplicial complexes is to find the lowest dimensional Euclidean space in which they can be embedded. We investigate this question for order complexes of posets. We show that order complexes of thick geometric…

We answer the following question posed by Lechuga: Given a simply-connected space $X$ with both $H_*(X,\qq)$ and $\pi_*(X)\otimes \qq$ being finite-dimensional, what is the computational complexity of an algorithm computing the cup-length…

We prove three optimal conformal geometric inequalities of Blatter type on the Klein bottle. These inequalities provide conformal lower bounds of the volume and involve lengths of homotopy classes of curves that are candidates to realize…

We use an alternative definition of topological complexity to show that the topological complexity of the mapping telescope of a sequence $X_1\rightarrow X_2\rightarrow X_3\rightarrow...$ is bounded above by $2max{TC(X_i); i=1,2,...}$.

We prove a bound for the geodesic diameter of a subset of the unit ball in $\mathbb{R}^n$ described by a fixed number of quadratic equations and inequalities, which is polynomial in $n$, whereas the known bound for general degree is…

We study the parameterized complexity of the following fundamental geometric problems with respect to the dimension $d$: i) Given $n$ points in $\Rd$, compute their minimum enclosing cylinder. ii) Given two $n$-point sets in $\Rd$, decide…

We present a viscosity approach to the min-max construction of closed geodesics on compact Riemannian manifolds of arbitrary dimension. We also construct counter-examples in dimension $1$ and $2$ to the $\varepsilon$-regularity in the…

Graphs that are critical (minimal excluded minors) for embeddability in surfaces are studied. In Part I, it was shown that graphs that are critical for embeddings into surfaces of Euler genus $k$ or for embeddings into nonorientable surface…

Let $f\colon M^{2n}\to\mathbb{R}^{2n+p}$ denote an isometric immersion of a Kaehler manifold of complex dimension $n\geq 2$ into Euclidean space with codimension $p$. If $2p\leq 2n-1$, we show that generic rank conditions on the second…

A minimal geodesic on a Riemannian manifold is a geodesic defined on $\mathbb{R}$ that lifts to a globally distance minimizing curve on the universal covering. Bangert proved that there is a lower bound for the number of geometrically…

The structure of graphs with a 2-vertex-cut that are critical with respect to the Euler genus is studied. A general theorem describing the building blocks is presented. These constituents, called hoppers and cascades, are classified for the…

It is shown that an embedded Lagrangian Klein bottle represents a non-trivial mod 2 homology class in a compact symplectic four-manifold $(X,\omega)$ with $c_1(X)\cdot[\omega]>0$. (In versions 1 and 2, the last assumption was missing. A…

We introduce the effectual topological complexity (ETC) of a $G$-space $X$. This is a $G$-equivariant homotopy invariant sitting in between the effective topological complexity of the pair $(X,G)$ and the (regular) topological complexity of…

The space of Gaussian measures on a Euclidean space is geodesically convex in the $L^2$-Wasserstein space. This space is a finite dimensional manifold since Gaussian measures are parameterized by means and covariance matrices. By…

In the present paper we prove, that if the geodesic flow of a metric G on the torus T is quadratically integrable, then the torus T isometrically covers a torus with a Liouville metric on it, and describe the set of quadratically integrable…

We study the minimal complexity of tilings of a plane with a given tile set. We note that every tile set admits either no tiling or some tiling with O(n) Kolmogorov complexity of its n-by-n squares. We construct tile sets for which this…

The extension complexity of a polytope measures its amenability to succinct representations via lifts. There are several versions of extension complexity, including linear, real semidefinite, and complex semidefinite. We focus on the last…

In this article, we prove a generalization of our previous result in [12]. In particular, we show that for an $n$-dimensional, simply connected Riemannian manifold with diameter $D$ and volume $V$. Suppose that $M$ admits a good cover…

For a polygonal domain with $h$ holes and a total of $n$ vertices, we present algorithms that compute the $L_1$ geodesic diameter in $O(n^2+h^4)$ time and the $L_1$ geodesic center in $O((n^4+n^2 h^4)\alpha(n))$ time, respectively, where…

Let $f\colon M^{2n}\to\mathbb{R}^{2n+4}$ be an isometric immersion of a Kaehler manifold of complex dimension $n\geq 5$ into Euclidean space with complex rank at least $5$ everywhere. Our main result is that, along each connected component…