Related papers: Covering Large Balls with Convex Sets in Spherical…
We derive an upper bound on the size of a ball such that the image of the ball under quadratic map is strongly convex and smooth. Our result is the best possible improvement of the analogous result by Polyak in the case of quadratic map. We…
This note shows that an interesting property arises when considering the relation between the hypersphere volumes at dimensions $n+1$ and $n$, if the hyperspheres circumscribe unitary hypercubes in $n+1$ and $n$ dimensions, respectively .…
High proved the following theorem. If the intersections of any two congruent copies of a plane convex body are centrally symmetric, then this body is a circle. In our paper we extend the theorem of High to spherical, Euclidean and…
We prove that the highest density of non-overlapping translates of a given centrally symmetric convex domain relative to its outer parallel domain of given outer radius is attained by a lattice packing in the Euclidean plane. This…
A new intrinsic volume metric is introduced for the class of convex bodies in $\mathbb{R}^n$. As an application, an inequality is proved for the asymptotic best approximation of the Euclidean unit ball by arbitrarily positioned polytopes…
The convex hull of a ball with an exterior point is called a spike (or cap). A union of finitely many spikes of a ball is called a spiky ball. If a spiky ball is convex, then we call it a cap body. In this note we upper bound the…
The space of shapes of a polyhedron with given total angles less than 2\pi at each of its n vertices has a Kaehler metric, locally isometric to complex hyperbolic space CH^{n-3}. The metric is not complete: collisions between vertices take…
We consider a spherical antiprism. It is a convex polyhedron with $2n$ vertices in the spherical space $\mathbb{S}^3$. This polyhedron has a group of symmetries $S_{2n}$ generated by a mirror-rotational symmetry of order $2n$, i.e. rotation…
For a convex body on the Euclidean unit sphere the spherical convex floating body is introduced. The asymptotic behavior of the volume difference of a spherical convex body and its spherical floating body is investigated. This gives rise to…
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 consider the problem of wrapping three-dimensional solid bodies with a given planar sheet of paper, where the paper may be folded or wrinkled but not stretched or torn. We propose a conjecture characterising the maximumvolume solid…
We present a survey article about the geometry of convex bodies on the $d$-dimensional sphere $S^d$. We concentrate on the results based on the notion of the width of a convex body $C \subset S^d$ determined by a supporting hemisphere of…
We define floating bodies in the class of $n$-dimensional ball-convex bodies. A right derivative of volume of these floating bodies leads to a surface area measure for ball-convex bodies which we call relative affine surface area. We show…
Given $N$ geodesic caps on the unit sphere in $\mathbb{R}^d$, and whose total normalized surface area sums to one, what is the maximal surface area their union can cover? In this work, we provide an asymptotically sharp upper bound for an…
The celebrated Dvoretzky theorem asserts that every $N$-dimensional convex body admits central sections of dimension $d = \Omega(\log N)$, which is nearly spherical. For many instances of convex bodies, typically unit balls with respect to…
We characterise the class of those Banach spaces in which every convex combination of slices of the unit ball intersects the unit sphere as the class of those spaces in which every convex combination of slices of the unit ball contains two…
We will show that for any $n\ge N$ points on the $N$-dimensional sphere $S^N$ there is a closed hemisphere which contains at least $\lfloor\frac{n+N+1}{2}\rfloor$ of these points. This bound is sharp and we will calculate the amount of sets…
In this paper, the following two theorems are proved: $(1)$ every spherical convex body $W$ of constant width $\Delta (W) \geq \frac{\pi}{2}$ may be covered by a disk of radius $\Delta(W) + \arcsin \left( \frac{2\sqrt{3}}{3} \cdot \cos…
In connection with an unsolved problem of Bang (1951) we give a lower bound for the sum of the base volumes of cylinders covering a d-dimensional convex body in terms of the relevant basic measures of the given convex body. As an…
We give a new and elementary proof that the number of elastic collisions of a finite number of balls in the Euclidean space is finite. We show that if there are $n$ balls of equal masses and radii 1, and at the time of a collision between…