Related papers: On asymptotic Lebesgue's universal covering proble…

The isodiametric inequality states that the Euclidean ball maximizes the volume among all convex bodies of a given diameter. We are motivated by a conjecture of Makai Jr.~on the reverse question: Every convex body has a linear image whose…

A set $M\subset\mathbb{R}$ is microscopic if for each $\varepsilon>0$ there is a sequence of intervals $(J_n)_{n\in\omega}$ covering $M$ and such that $|J_n|\leq \varepsilon^{n+1}$ for each $n\in\omega$. We show that there is a microscopic…

Many classical problems in convex geometry can be cast as optimization problems under certain containment conditions. The arguably best-understood example is volume-maximization of convex bodies contained in other convex bodies, where the…

For n >= 2 a construction is given for a large family of compact convex sets K and L in n-dimensional Euclidean space such that the orthogonal projection L_u onto the subspace u^\perp contains a translate of the corresponding projection K_u…

Leighton's graph covering theorem says that two finite graphs with a common cover have a common finite cover. We present a new proof of this using groupoids, and use this as a model to prove two generalisations of the theorem. The first…

Let $B_n$ be the $n$-dimensional unit ball given by the inequality $\|x\|\leq 1$, where $\|x\|$ is the standard Euclid norm in ${\mathbb R}^n$. For an $n$-dimensional nondegenerate simplex $S$, we denote by $E$ the ellipsoid of minimum…

We discuss generalizations of the well-known theorem of Hilbert that there is no complete isometric immersion of the hyperbolic plane into Euclidean 3-space. We show that this problem is expressed very naturally as the question of the…

We establish estimates for the asymptotic best approximation of the Euclidean unit ball by polytopes under a notion of distance induced by the intrinsic volumes. We also introduce a notion of distance between convex bodies that is induced…

Thurston's circle packing approximation of the Riemann Mapping (proven to give the Riemann Mapping in the limit by Rodin-Sullivan) is largely based on the theorem that any topological disk with a circle packing metric can be deformed into a…

In \cite{LiWang2001complete1,LiWang2001complete2}, Li-Wang proved a splitting theorem for an n-dimensional Riemannian manifold with $Ric\geqslant -(n-1)$ and the bottom of spectrum $\lambda_0(M)=\frac{(n-1)^2}{4}$. For an n-dimensional…

Let $P_{n}$ be a set of $n$ points, including the origin, in the unit square $U = [0,1]^2$. We consider the problem of constructing $n$ axis-parallel and mutually disjoint rectangles inside $U$ such that the bottom-left corner of each…

Polytopes are the basic finite data structures for convex sets: they appear as feasible regions in linear optimization, as geometric summaries in algorithms, and as random objects in stochastic geometry. A natural geometric question is…

We find an optimal upper bound on the volume of the John ellipsoid of a $k$-dimensional section of the $n$-dimensional cube, and an optimal lower bound on the volume of the L\"owner ellipsoid of a projection of the $n$-dimensional…

Theorem A. Let $M^n$ denote a closed Riemannian manifold with nonpositive sectional curvature and let $\tilde M^n$ be the universal cover of $M^n$ with the lifted metric. Suppose that the universal cover $\tilde M^n$ contains no totally…

We prove a generalized isoperimetric inequality for a domain diffeomorphic to a sphere that replaces filling volume with $k$-dilation. Suppose $U$ is an open set in $\mathbb{R}^n$ diffeomorphic to a Euclidean $n$-ball. We show that in…

Theoretical background is provided towards the mathematical foundation of the minimum enclosing ball problem. This problem concerns the determination of the unique spherical surface of smallest radius enclosing a given bounded set in the…

We strengthen, in various directions, the theorem of Garnett that every $\sigma$-compact, completely regular space $X$ occurs as a Gleason part for some uniform algebra. In particular, we show that the uniform algebra can always be chosen…

Suppose A is a finite set equipped with a probability measure P and let M be a ``mass'' function on A. We give a probabilistic characterization of the most efficient way in which A^n can be almost-covered using spheres of a fixed radius. An…

A generalization of the Lebesgue number lemma is obtained. It is proved that, if each countably infinite locally finite open cover of a chainable metric space $X$ has a Lebesgue number, then $X$ is totally bounded. A property of metric…

It is a well known fact that in $\mathbb{R}^n$ a subset of minimal perimeter $L$ among all sets of a given volume is also a set of maximal volume among all sets of the same perimeter $L$. This is called the reciprocity principle for…