All Questions
Tagged with mg.metric-geometry convex-geometry
328 questions
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On $N$-partition of some common subsets $\Omega\subset\mathbb R^d$
Let $\Omega\subset\mathbb R^d$ be compact and convex, and denote by $\ell$ the normalised Lebesgue measure such that $\ell(\Omega)=1$. Let $N$ be an arbitrary but fixed integer.
In this post we set $d=...
- 1,399
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40
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Converse of Scherk–Segre theorem on the number of vertices of a convex space curve
It is well known that any smooth simple closed convex curve $\gamma$ in $\mathbb{R}^{3}$ that meets no plane in more than 4 points has exactly 4 vertices, i.e., points of vanishing torsion; here "...
- 975
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1
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168
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Ratio of inscribed/circumscribed ellipsoids: geometrical proof?
Let $K$ be a convex subset of ${\mathbb R}^n$, with non-void interior. The Löwner-John theorem states that there are a minimal volume ellipsoid $\cal E$ containing $K$, a maximal one $\cal F$ ...
- 52.3k
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37
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Metric entropy of an ellipsoid
Let $B^d_2$ denote the unit ball of $\ell_2^d$ and let $T$ be an invertible linear map.
Consider the function
$$
H(T) := \log M(TB_2^d, B_2^d),
$$
which is the packing entropy for $TB_2^d$ by $B_2^d$....
- 410
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37
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Constructing a minimum-volume outer approximation polytope with fewer facets
I am tackling the following problem:
Given a set of points $D \in \mathbb{R}^d$ and their convex hull, represented with $n$ facets, I want to construct a convex polytope $P$ with at most $m<n$ ...
- 101
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1
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347
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Is a ball the hardest body to approximate by polytopes (in the Banach–Mazur metric)?
$\DeclareMathOperator\conv{conv}\DeclareMathOperator\Vol{Vol}$In the paper "An extremal property of the hypersphere" by Macbeath, the following functionals were introduced (here $n$ is fixed,...
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Is it necessarily true that the maximal section of a centrally symmetric convex body is always bigger than its minimal projection?
I hope everyone is doing well.
Let $K \subset \mathbb{R}^n$ be a centrally symmetric convex body $(K = -K)$. Denote by $K \mid H$ the orthogonal projection of $K$ onto $H$, where $H$ is an $n - 1$ ...
- 83
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67
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Projection of a gaussian random vector onto a convex body
Let $K \subset \mathbb{R}^n$ denote a convex body. Let $\Pi_K$ denote the projection onto $K$,
$$
\Pi_K(y) = \mathrm{arg\,min}_{x \in K} \|y - x\|,
$$
where $\|\cdot\|$ denotes the usual Euclidean ...
- 410
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21
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Largest inscribed parallelepiped of the convex set defined by partial sum of Fourier series
Let $\mathcal{X}$ be the set consisting of all $(2n+1)$-dimensional real vectors $\mathbf{x}=\left( a_0,a_1,\ldots,a_n,b_1,\ldots,b_n\right)^{\intercal}$ satisfying
$$
\left| f_{\mathbf{x}}(t) \right|...
- 550
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316
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Sandwiching ellipses between planar convex bodies
Let $K$ and $L$ be planar convex bodies which are not ellipses. Does there exist an affine image $K'$ of $K$ such that
$K' \subset L$
No ellipse $E$ satisfies $K' \subset E \subset L$
I am also ...
- 4,653
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71
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Is the circumcenter Lipschitz on large convex sets in hyperbolic space?
Given a uniquely geodesic metric space $X$, let $\mathcal K(X)$ denote the metric space of compact, convex subsets of $X$ equipped with the Hausdorff distance. Given $K \in \mathcal K(X)$, let $c(K)$ ...
- 498
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70
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Lipschitz continuity of orthogonal projection with respect to the Hausdorff distance
Let $x_0 \in \mathbb R^n$, and let $\mathcal K$ denote the set of compact convex subsets of $\mathbb R^n$ equipped with the Hausdorff metric. Consider the map $f: K \mapsto \Pi_K x_0$, where $\Pi_K$ ...
- 498
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132
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Can I find $n$ points on the boundary of an $n$-dimensional ball with certain properties?
My problem is the following: I want to construct $n$ rays all starting at a point $v$ that is not in the $n$-dimensional ball around $0$ such that the following is true:
The $n$-dimensional ball is a ...
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63
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Convex planar regions such that every boundary point has a 'fair bisector' passing thru it
We add a little to On 'fair bisectors' of planar convex regions and A claim on the concurrency of area bisectors of planar convex regions .
A fair bisector of a planar convex region is a line ...
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82
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On 'Bisecting sections' of 3D convex bodies
Following shadows and planar sections, we ask about bisecting sections. This post also continues Convex planar regions with all area bisectors having equal length and A claim on the concurrency of ...
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136
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If all max area planar sections of a solid are centrally symmetric, will the solid as whole be centrally symmetric?
It is known that every planar section of an ellipsoid is an ellipse - a centrally symmetric planar figure.
Are there convex solids other than ellipsoids with the property that all its planar sections ...
- 5,979
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42
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On a pair of solids with both corresponding maximal planar sections and shadows having equal area
This post pulls together Are two convex solids with all corresponding shadows equal in area congruent? and
What can be said about 2 convex solids with corresponding maximal planar sections having ...
- 5,979
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What can be said about 2 convex solids with corresponding maximal planar sections having equal area?
This post follows Are two convex solids with all corresponding shadows equal in area congruent?
Every convex 3D body has planar sections with normals in any given direction. We consider the maximum ...
- 5,979
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1
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302
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Are two convex solids with all corresponding shadows equal in area congruent?
By shadow we mean the orthogonal projection of a convex 3D body P onto a 2D plane, for example, the shadow on the xy-plane, with P above (z>0) that plane and the light at L=(0,0,+∞). P an be freely ...
- 5,979
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404
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Estimating shortest paths in planar drawings of graphs
Consider a drawing (in $\mathbb{R}^2$) of a planar graph. (The drawing is given, contrarily to the common setup in graph theory where we are seeking to build a drawing with specific properties.)
For ...
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4
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2
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154
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Pushing a convex cone and equidistants
Let $K$ be a closed convex cone in an n-dimensional Euclidean space.
Suppose $K$ has non-empty interior. For $t > 0$
form the subcone $K_t$ consisting of all points in $K$ which lie a ...
- 8,336
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44
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On area bisectors and perimeter bisectors of planar convex regions
We try to proceed from A claim on the concurrency of area bisectors of planar convex regions
Definitions: Given a planar convex region C, an area bisector of C is any line segment that partitions C ...
- 5,979
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86
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Busemann-Petty type problems on complex vector spaces [closed]
We recently published an article on Busemann-Petty type problems (see https://arxiv.org/abs/2404.05630). As we experienced several times that as soon as an article is published, no updates/corrections/...
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95
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Distance between two convex sets
Setting
If $A$ an $B$ are two symmetric matrices, we denote by $A >B$ when the matrice $A-B$ is definite positive.
In $\left(\mathbb{R}^{*}_{+} \right)^4$, consider the convex set $$ \Lambda = \...
- 125
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413
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How many unit cubes are needed to 'hide' a unit cube fully in 3D?
Question: What is the smallest number of nonoverlapping unit cubes that can hide a unit cube C - in the sense that every ray emanating from the boundary of C meets the interior or the boundary of one ...
- 5,979
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84
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Simple convergence of convex compact set implies Hausdorff convergence
I am wondering about the following :
In $\mathbb{R}^n$, suppose you are given compact convex bodies $\left\{ C_k : k \geq 1 \right\}$ and $C$, such that for every $x \in \mathbb{R}^n$ $$ \mathbb{1}_{...
- 125
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40
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Polyhedra inscribed in a sphere with mutually non-congruent, equal area faces
Two constrained versions of the main question given in this post: Polyhedrons with mutually non-congruent faces, all of equal area. An earlier post that could be related: Cutting a spherical surface ...
- 5,979
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98
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To place copies of a planar convex region such that number of 'contacts' among them is maximized
A contact between two planar convex regions obviously happens either along a line segment or at a single point.
Question: Given a planar convex region $C$ and a number $n$, we need to lay out $n$ ...
- 5,979
4
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197
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What are the measure of the volume and boundary (and other quermaß measures) of the positive semidefinite matrices?
Let $E$ be the real vector space of $n\times n$ real symmetric (resp. complex Hermitian) matrices, and $E_1$ those with trace $1$. Endow $E$ with the bilinear (resp. sesquilinear) form given by $(P,Q)...
- 32.5k
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49
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Which planar convex region with specified area and perimeter maximizes/minimizes Moment of Inertia?
By moment of inertia of a planar convex region C, here we mean its moment of inertia about an axis passing through the center of mass of C and perpendicular to the plane of C.
Question: For specified ...
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139
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Characterization of convexity by connectedness of hyperplane sections
Let $S$ be a smooth closed connected embedded hypersurface in $\mathbb R^n$.
Is it true that $S$ is convex, i.e. is a boundary of a convex set, if and only if any section of $S$ by a hyperplane is ...
- 2,934
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208
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Reference request: Carathéodory-type theorem for convex hulls of closed sets
I'm looking for a reference for the following theorem.
Theorem Let $X$ be a closed subset of $\mathbb{R}^N$, and let $a$ be a point of its convex hull $\operatorname{conv}(X)$. Then there exist ...
- 27.7k
2
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1
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383
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Geometry in $\mathbb{R}^n$: angle between projections of a rectangle
Consider a hyper rectangle $R$ in $\mathbb{R}^n$ defined by $|x_i|\leq M_i$ for all $i\leq n$.
Consider a linear affine subspace $L$ of dimension $1\leq k <n$ such that $L\cap R\neq \emptyset$.
For ...
- 21
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1
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186
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Convex hull of 3 points in Cartan-Hadamard manifolds
Can the convex hull of $3$ points in a Cartan-Hadamard manifold be smooth?
A Cartan-Hadamard manifold $M$ is a complete simply connected manifold with nonpositive curvature (so it includes the ...
- 7,149
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1
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107
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To find the convex planar region minimizing diameter when area and perimeter are given
The basic question is to find that planar convex region for which diameter is a minimum when area and perimeter are specified.
A partial answer is given here: http://nandacumar.blogspot.com/2012/11/...
- 5,979
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107
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Planar sections of convex sets in Cartan-Hadamard manifolds
Let $X$ be a convex set in Euclidean space $\mathbf{R}^n$ and $p\in\mathbf{R}^n$ be a fixed point. Then any plane $\Pi$ passing through $p$ intersects $X$ in a convex set. Conversely, this property ...
- 7,149
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John-type theorems: trading structure for accuracy?
Given two symmetric convex bodies $B, B'$ in ${\bf R}^d$, define the Banach-Mazur distance $d(B,B')$ between them to be the least constant $\tau \geq 1$ such that
$$ B \subset TB' \subset \tau B$$
for ...
- 114k
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51
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Estimating the Hausdorff distance of parallel facets of convex polytopes
Background
Let $\mathcal{K}_P^n$ denote the class of open, convex, $n$-dimensional polytopes in $\mathbb{R}^n$ containing the origin. For each $K\in \mathcal{K}_P^n$, its gauge function $f_*:\mathbb{R}...
- 21
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56
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Zero flux along lines
I am considering the $L^1$ ball in $\mathbb{R}^d$, and a conservative vector field $V$ on it, which arises as the gradient of a bounded, almost-everywhere Lipchitz-function. Denoting by $e_i$ as the i’...
- 232
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46
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Complexity for determining whether a given metric space is hyperconvex?
Suppose I am given a finite metric space as a distance matrix. What is the complexity of determining whether this metric space is hyperconvex?
Definition: A metric space is said to be hyperconvex if ...
1
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1
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75
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When do the centers of mass of a uniform convex planar region as a whole and of its boundary alone coincide?
Given a uniform planar convex region C, let us consider 2 centers of mass - the center of mass of the region as a whole and the center of mass of its boundary alone (assuming its boundary to have ...
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132
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Planar convex region maximizing the difference in 'orientation' between its smallest containing rectangle and largest contained rectangle
We say a rectangle has orientation $\theta$ if the vector from its center to the middle of its shortest side (parallel to the longest side) has some angle $\theta$ with X axis.
Consider a planar ...
- 5,979
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76
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A claim on planar sections of 3D convex bodies
Ref: More on shadows of 3D convex bodies,
Shadows and planar sections of polyhedra
Given a 3D convex body C, we define a maximal area (perimeter) section of C with respect to any specified direction $...
- 5,979
2
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0
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114
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More on shadows of 3D convex bodies
Ref: Shadows and planar sections of polyhedra
By shadow we mean the orthogonal projection of a convex 3D body C onto a 2D plane, for example, the shadow on the xy-plane, with C above (z>0) that ...
- 5,979
1
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52
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On families of lines that cut the boundary of a planar convex region in a specified ratio
We proceed from A claim on the concurrency of area bisectors of planar convex regions
This question is somewhat broad.
Background: 'Mathematical Omnibus' by Fuchs and Tabachnikov, Lecture 11 describes ...
- 5,979
1
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2
answers
157
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A claim on the concurrency of area bisectors of planar convex regions
We add a little bit to On 'fair bisectors' of planar convex regions and Bisectors and partitioning lines for convex regions defined with respect to the moment of inertia
Definitions: Given a ...
- 5,979
1
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0
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83
views
Closed form volumes for intersecting modified cylinders
This question is somewhat related to the question Intersecting cylinders, but where the cylinders are now modified to orbifolds in the hypercube with singularities occurring at the vertices of the ...
- 383
6
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2
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539
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Conditions for including cones
Consider $N$ $n$-dimensional vectors, where the angle between any two vectors is acute and their starting point is at the origin. Can we rotate these vectors together so that the coordinate components ...
- 61
4
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0
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52
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Quantifying error in the reconstruction of convex polytopes from moments
The problem of reconstructing a geometric object from its moments is of interest in a variety of fields. In the paper The Inverse Moment Problem for Convex Polytopes, the authors show that a convex ...
- 181
1
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0
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61
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Fitting a convex polytope with 𝑛 facets between two nested spheres
This is related to a research problem that is interested in approximation of spheres by convex polytopes.
Let $C_r$ and $C_R$ be two spheres in $\mathbb R^d$ of radius $r$ and $R$, respectively, where ...