arXiv subjects
Publications and source records attributed to Marek Lassak.
Denote by $S^2$ the two-dimensional sphere. A spherical convex body on $S^2$ which does not properly contain a spherical convex body of the same spherical thickness is called a reduced body. We give three characterizations of reducedness of spherical convex odd-gons on $S^2$. Analogous characterizations hold true also in the Euclidean and hyperbolic planes.
We prove a theorem on the relationships between the lengths of sides of a spherical quadrilateral with three right angles. They are analogous to the relationships in the Lambert quadrilateral in the hyperbolic plane. We apply this theorem in the proof of our second theorem that if $C$ is a two-dimensional spherical convex body of diameter $\delta \in (\frac{1}{2}\pi,\pi)$, then the diameter of the set of extreme points of $C$ is at least $2 \arccos \big(\frac{1}{4}(\cos \delta + \sqrt {\cos^2 \delta +8})\big)$. This estimate cannot be improved.
For a hyperplane $H$ supporting a convex body $C$ in the hyperbolic space $\mathbb{H}^d$ we define the width of $C$ determined by $H$ as the distance between $H$ and a most distant ultraparallel hyperplane supporting $C$. The minimum width of $C$ over all supporting $H$ is called the thickness $Δ(C)$ of $C$. A convex body $R \subset \mathbb{H}^d$ is said to be reduced if $Δ(Z) < Δ(R)$ for every convex body $Z$ properly contained in $R$. We describe a class of reduced polygons in $\mathbb{H}^2$ and present some properties of them. In particular, we estimate their diameters in terms of their thicknesses.
We consider the variant of the Banach-Mazur distance $δ_{BM}^{\rm cen} (C, D)$ of two convex bodies $C, D$ of $E^d$ with the additional requirement that the centroids of them coincide. We prove that $δ_{BM}^{\rm cen} (C, D) \leq \frac{69}{17}$ for every $C, D$ of $E^2$.
For a hyperplane $H$ supporting a convex body $C$ in the hyperbolic space $\mathbb{H}^d$ we define the width of $C$ determined by $H$ as the distance between $H$ and a most distant ultraparallel hyperplane supporting $C$. The thickness (i.e., the minimum width) of $C$ is denoted by $Δ(C)$. A convex body $R \subset \mathbb{H}^d$ is called reduced if for every body $Z \subsetneq R$ we have $Δ(Z) < Δ(R)$. We show that for any extreme point $e$ of a reduced body $R \subset \mathbb{H}^d$ there exists a supporting hyperplane $H$ of $R$ which passes through $e$ or its equidistant surface supporting $R$ passes through $e$. Bodies of constant width in $\mathbb{H}^d$ are defined as bodies whose all widths are equal. We prove that every complete body in $\mathbb{H}^d$ is a body of constant width.
For every hyperplane $H$ supporting a convex body $C$ in the hyperbolic space $\mathbb{H}^d$ we define the width of $C$ determined by $H$ as the distance between $H$ and a most distant ultraparallel hyperplane supporting $C$. We prove that if $\width_H (C) = Δ(C)$ and if there exists a unique most distant point $j \in C$ from $H$, then the projection of $j$ onto $H$ belongs to $H \cap C$. We verify that the diameter of $C$ equals to the maximum width of $C$. We define bodies of constant width in $\mathbb{H}^d$ in the standard way as bodies whose all widths are equal. We show that every body of constant width is strictly convex. The minimum width of $C$ over all supporting $H$ is called the thickness $Δ(C)$ of $C$. A convex body $R \subset \mathbb{H}^d$ is said to be reduced if $Δ(Z) < Δ(R)$ for every convex body $Z$ properly contained in $R$. We show that regular tetrahedra in $\mathbb{H}^3$ are not reduced. Similarly as in the Euclidean and spherical spaces, we introduce complete bodies and bodies of constant diameter in $\mathbb{H}^d$. We show that every body of constant width $δ$ is a body of constant diameter $δ$ and a complete body of diameter $δ$. Moreover, the two last conditions are equivalent.
It is well known that any planar convex body $A$ permits to inscribe an affine-regular hexagon $H_A$. We prove that the centroid of $A$ belongs to the homothetic image of $H_A$ with ratio $\frac{4}{21}$ and the center in the center of $H_A$. This ratio cannot be enlarged.
Let $C$ and $D$ be convex bodies in the Euclidean space $E^d$. We define the centroid Banach-Mazur distance $δ_{BM}^{\rm cen} (C, D)$ similarly to the classic Banach-Mazur distance $δ_{BM} (C, D)$, but with the extra requirement that the centroids of $C$ and an affine image of $D$ coincide. We prove that for the parallelogram $P$ and the triangle $T$ in $E^2$ we have $δ_{BM}^{\rm cen} (P, T) = \frac{5}{2}$.
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 $C$. Important tools are the lunes containing $C$. The supporting hemispheres take over the role of the supporting half-spaces of a convex body in Euclidean space, and lunes the role of strips. Also essential is the notion of thickness of $C$, i.e, its minimum width. In particular, we describe properties of reduced spherical convex bodies and spherical bodies of constant width. The last notion coincides with the notions of complete bodies and bodies of constant diameter on $S^d$. The reminded and commented here results concern mostly the width, thickness, diameter, perimeter, area and extreme points of spherical convex bodies, reduced bodies and bodies of constant width.
We present a spherical version of the theorem of Blaschke that every body of constant width $w < \fracπ{2}$ can be approximated as well as we wish in the sense of the Hausdorff distance by a body of constant width $w$ whose boundary consists only of pieces of circles of radius $w$. This is a special case of our theorem about approximation of spherical reduced bodies.
We generalize Monge's theorem for $n+1$ pairwise homothetic sets (in particular convex bodies) in $E^n$ in place of three disks in $E^2$. We also present a version for $n+1$ independent points of $E^n$. It also includes the reverse statement. Moreover, we give an analogon of Monge's theorem for the $n$-dimensional sphere and hyperboloid model of the hyperbolic space.
Similarly to the classic notion in $E^d$, a subset of a positive diameter below $\fracπ{2}$ of a hemisphere of the sphere $S^d$ is called complete, provided adding any extra point increases its diameter. Complete sets are convex bodies on $S^d$. Our main theorem says that on $S^d$ complete bodies of diameter $δ$ coincide with bodies of constant width $δ$.
We show that the Banach-Mazur distance between the parallelogram and the affine-regular hexagon is $\frac{3}{2}$ and we conclude that the diameter of the family of centrally-symmetric planar convex bodies is just $\frac{3}{2}$. A proof of this fact does not seem to be published earlier. Asplund announced this without a proof in his paper proving that the Banach-Mazur distance of any planar centrally-symmetric bodies is at most $\frac{3}{2}$. Analogously, we deal with the Banach-Mazur distances between the parallelogram and the remaining affine-regular even-gons.
We present some relationships between the diameter, width and thickness of a reduced convex body on the $d$-dimensional sphere. We apply the obtained properties to recognize if a Wulff shape in the Euclidean $d$-space is self-dual.
{\bf Abstract.} Let $D$ be a convex body of diameter $δ$, where $0 < δ< \fracπ{2}$, on the $d$-dimensional sphere. We prove that $D$ is of constant diameter $δ$ if and only if it is of constant width $δ$ in the following two cases. The first case is when $D$ is smooth. The second case is when $d=2$.
The intersection $L$ of two different non-opposite hemispheres of the unit sphere $S^2$ is called a lune. By $Δ(L)$ we denote the distance of the centers of the semicircles bounding $L$. By the thickness $Δ(C)$ of a convex body $C \subset S^2$ we mean the minimal value of $Δ(L)$ over all lunes $L \supset C$. We call a convex body $R\subset S^2$ reduced provided $Δ(Z) < Δ(R)$ for every convex body $Z$ being a proper subset of $R$. Our aim is to estimate the diameter of $R$, where $Δ(R) < \fracπ{2}$, in terms of its thickness.
The intersection $L$ of two different non-opposite hemispheres $G$ and $H$ of a $d$-dimensional sphere $S^d$ is called a lune. By the thickness of $L$ we mean the distance of the centers of the $(d-1)$-dimensional hemispheres bounding $L$. For a hemisphere $G$ supporting a %spherical convex body $C \subset S^d$ we define ${\rm width}_G(C)$ as the thickness of the narrowest lune or lunes of the form $G \cap H$ containing $C$. If ${\rm width}_G(C) =w$ for every hemisphere $G$ supporting $C$, we say that $C$ is a body of constant width $w$. We present properties of these bodies. In particular, we prove that the diameter of any spherical body $C$ of constant width $w$ on $S^d$ is $w$, and that if $w < \fracπ{2}$, then $C$ is strictly convex. Moreover, we are checking when spherical bodies of constant width and constant diameter coincide.
Denote by ${\mathcal K}^d$ the family of convex bodies in $E^d$ and by $w(C)$ the minimal width of $C \in {\mathcal K}^d$. We ask for the greatest number $Λ_n ({\mathcal K}^d)$ such that every $C \in {\mathcal K}^d$ contains a polytope $P$ with at most $n$ vertices for which $Λ_n ({\mathcal K}^d) \leq \frac{w(P)}{w(C)}$. We give a lower estimate of $Λ_n ({\mathcal K}^d)$ for $n \geq 2d$ based on estimates of the smallest radius of $\big\lfloor {\frac{n}{2}} \big\rfloor$ antipodal pairs of spherical caps that cover the unit sphere of $E^d$. We show that $Λ_3 ({\mathcal K}^2) \geq {\frac 1 2}(3- \sqrt 3)$, and $Λ_n ({\mathcal K}^2) \geq \cos {\frac π{2 \lfloor {n/2} \rfloor}}$ for every $n \geq 4$. We also consider the dual question of estimating the smallest number $Δ_n ({\mathcal K}^d)$ such that every $C \in {\mathcal K}^d$ there exists a polytope $P \supset C$ with at most $n$ facets for which $\frac{{\rm diam}(P)}{{\rm diam}(C)} \leq Δ_n ({\mathcal K}^d)$. We give an upper bound of $Δ_n ({\mathcal K}^d)$ for $n \geq 2d$. In particular, $Δ_n ({\mathcal K}^2) \leq 1/ \cos {\frac π{2 \lfloor {n/2} \rfloor}}$ for $n \geq 4$.