All Questions
71 questions
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Helly's number from biconvex functions
Helly's Theorem states the following. Suppose $X_1,X_2,...,X_N$ are convex sets in $\mathbb{R}^d$, such that for any index-set $I$ with $|I| \leq h(d) := d+1$, we have $\bigcap_{i \in I} X_i \neq \...
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On convex polygons contained in convex polygons
In what follows '$n$-gon' stands for '$n$-vertex polygonal region'.
Question: Given a convex $n$-gon $C$, find the smallest convex region $R$ such that $C$ is the smallest $n$-gon that contains it.
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2
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153
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Smallest triangles that contain 2D convex regions with reflection symmetry
Given any 2D convex region $C$ with a mirror symmetry. Two pairs of questions:
We need to find the smallest area (likewise, smallest perimeter) triangle that contains $C$. Is it sufficient to only ...
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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 $n$ ...
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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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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 ...
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59
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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 ...
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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 ...
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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 ...
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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 ...
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Possible extensions of the perpendicular axes theorem for moment of inertia
This post continues on Moment of inertia from Bisectors and partitioning lines for convex regions defined with respect to the moment of inertia.
The perpendicular axis theorem states that the moment ...
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More on triangles inscribed in convex regions with one vertex fixed
We add a bit to On maximum perimeter triangles inscribed in convex regions with one vertex fixed. Let C be a convex planar region and P a point on its boundary.
Are there convex shapes C other than (...
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111
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Maximizing the minimum curvature of a convex shape with a given volume in higher dimensions
Given any $d$-dimensional convex shape $S$ in the Euclidean space with $d\gg 1$, let $K_{\min}(S)$ be the minimum value of the Gaussian curvature of its boundary.
Question: What is the maximum value $...
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76
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Convex planar regions with optimal average 'centralness' and 'depth'
For a planar convex region $C$ and an interior point $P$ we define:
the centralness ratio at $P$ is
$$\min\left(\frac{\text{shorter portion of }\chi}{\text{longer portion of }\chi}:\chi\text{ is a ...
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153
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Is there a polynomial expression for the volume of the following set?
Denote the unit $\ell_2$ ball in $\mathbb{R}^n$ as $\mathcal{B}_n$. It is widely kown that for a convex body $\mathcal{K}\subseteq \mathbb{R}^n$, the $n$-dimensional volume of the parallel body $\...
- 550
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Shadows and planar sections of polyhedra – 2
This post continues Shadows and planar sections of polyhedra and On planar sections of 3D convex bodies
Shadows and planar sections of polyhedra gives an example demonstrating that shadows (orthogonal ...
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124
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Number of lattice points in a structural symmetric convex body
Let $f$ is a convex symmetric function on the interval $[-a,a]$, i.e., $f(-x)=f(x)$ for $\forall \, x\in [-a,a]$. Then we consider a $n$-dimensional convex body in Euclidean space
\begin{equation}
\...
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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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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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The closest ellipse and circle to a given triangle - 2
We add a little more to The closest ellipse to a given triangle.
The above linked discussion used the Hausdorff distance to quantify how close two planar convex regions are.
In an earlier post - ...
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On 'axiality' of planar convex regions
Axiality has been studied under a definition given here: https://en.wikipedia.org/wiki/Axiality_(geometry)
Consider an alternative definition of axiality as follows: For a convex region C, consider a ...
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