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Conjecture on the increasing efficiency of the shortest minimum-link polygonal chains covering any grids of the form $\{0,1,2\}^k$ as $k$ grows

From the well-known Nine dots problem, we know that we need a polygonal chain with at least $4$ edges to connect the $9$ points of the planar grid $G_{3,2}:=\{\{0, 1, 2\} \times \{0, 1, 2\}\} \subset \...
Marco Ripà's user avatar
  • 1,451
1 vote
0 answers
111 views

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 $...
Penelope Benenati's user avatar
1 vote
0 answers
124 views

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} \...
RyanChan's user avatar
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1 vote
0 answers
81 views

Constructive way to optimally cover a compact subset of Euclidean space

Let, $(X,d)$ be a simply connected compact subset of $\mathbb{R}^d$ with non-empty interiorn, let $d$ denote the Euclidean metric, and let $\varepsilon>0$. Is there a way to iteratively select ...
ABIM's user avatar
  • 5,405
0 votes
1 answer
229 views

Is this bounded?

May be better to ask for help here. Let $v_{1}$, $v_{2}$, $\ldots$, $v_{m}$ be the vertices of a convex polygon in the plane and $v_{m+1}$ be a vertex in the interior of the convex polygon. Connect ...
Palt's user avatar
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0 answers
176 views

How to find a configuration of lines

In $\mathbb{R}^3$, can anyone help find a configuration of 5 lines such that the minimum of the smallest semi-axis lengths of the ellipsoid $ \mathbf{x}^T \mathbf{A} \mathbf{x} = 1 $, where $\mathbf{A}...
Don's user avatar
  • 61
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82 views

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 ...
Nandakumar R's user avatar
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49 views

When can a compact metric space be covered by finitely many nearly-disjoint closed and convex sets?

This question is a follow-up of the following negative question. Let $(X,d)$ be a (non-empty) compact metric space. More generally than in the first post, I'll call a set of non-empty subsets $C_1,\...
ABIM's user avatar
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