Questions tagged [convex-geometry]
A branch of geometry dealing with convex sets and functions. Polytopes, convex bodies, discrete geometry, linear programming, antimatroids, ...
1,006
questions
86
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Continuous maps which send intervals of $\mathbb{R}$ to convex subsets of $\mathbb{R}^2$
Let $f : \mathbb{R} \longrightarrow \mathbb{R}^2$ be a continuous map which sends any interval $I \subseteq \mathbb{R}$ to a convex subset $f(I)$ of $\mathbb{R}^2$. Is it true that there must be a ...
63
votes
6
answers
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Shortest closed curve to inspect a sphere
Let $S$ be a sphere in $\mathbb{R}^3$. Let $C$ be a closed curve in $\mathbb{R}^3$ disjoint from and
exterior to $S$
which has the property that every point $x$ on $S$ is visible to some point $y$ of $...
45
votes
4
answers
5k
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Polynomial roots and convexity
A couple of years ago, I came up with the following question, to which I have no answer to this day. I have asked a few people about this, most of my teachers and some friends, but no one had ever ...
44
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11
answers
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Algorithm for finding the volume of a convex polytope
It's easy to find the area of a convex polygon by division into triangles, but what is the optimal way of finding the volume of higher-dimensional convex bodies? I tried a few methods for dividing ...
41
votes
6
answers
2k
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Approximating a convex disk by an ellipse
For every convex compact set $K$ of area $1$ in $\mathbb{R}^2$, among all ellipses of area $1$ there exists an ellipse $E$ such that the area of the symmetric difference between $K$ and $E$ is ...
39
votes
2
answers
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Abstract definition of convex set
I'd like to formulate an abstract definition of convex sets: a set $K$ is convex if it is endowed with a ternary operation $K\times[0,1]\times K\to K$, written $(x:t:y)$, satisfying axioms
$(x:0:y)=(...
38
votes
7
answers
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Shortest path connecting two opposite points on a cube
Is it true, that a path connecting two opposite points (i.e. such that the segment joining them passes through the centre of mass of the cube) on the surface of the $d$-dimensional unit cube (with $d&...
38
votes
2
answers
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How to make a sandwich from just one piece of bread?
I don't know how to go about such questions. It's not exactly my area, so maybe it is stupid, but curiosity is winning.
So I have a piece of bread $P$ of a really non-regular shape (let's make it ...
37
votes
0
answers
1k
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Converse of the Archimedean property of the sphere
In his remarkable book On the Sphere and Cylinder, where he came tantalizingly close to discovering calculus, Archimedes showed that the area of the portion of the sphere contained between a pair of ...
36
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0
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Two-convexity ⇒ Lefschetz?
Assume that
$\Omega$ is an open simply connected set in $\mathbb R^n$
(two-convexity) if 3 faces of a 3-simplex belong to $\Omega$ then whole simplex in $\Omega$.
Is it true that any component of ...
35
votes
1
answer
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Which polygons can be turned inside out by a smooth deformation?
Take a non-degenerate polygon with side lengths $\{a_1,\dots,a_n\}$ in a convex configuration. What is the condition on the $a_i$'s so that the polygon can be turned inside out by a continuous motion ...
34
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6
answers
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How to explain the concentration-of-measure phenomenon intuitively?
One way to phrase the
"concentration-of-measure"
phenomenon is that,
for a Euclidean sphere $S^d$ in $d$ dimensions, for large $d$,
"most of the mass is close to the equator, for any equator."1
Q. ...
33
votes
3
answers
2k
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Polar body of a convex body that avoids a lattice
Let $K \subset {\bf R}^d$ be a symmetric convex body (an open bounded convex neighbourhood of the origin with $K = -K$) with the property that $K + {\bf Z}^d \neq {\bf R}^d$, i.e. the projection of $K$...
31
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2
answers
2k
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The logic of convex sets
Let me start with Helly's theorem: Let $A_1$, $A_2$, ..., $A_{n+2}$ be $n+2$ convex subsets of $\mathbb R^n$. If any $n+1$ of these subsets intersect (this means: have nonempty intersection), the so ...
28
votes
2
answers
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Can we always shift two disjoint convex bodies a little bit to decrease the volume of their convex hull?
Let $K,L\subset\mathbb R^d$ be two disjoint compact convex sets with non-empty interiors. Can $x=0$ be a point of local minimum for the function $F(x)=\text{vol}_d(\text{conv(K,L+x))}$?
I was asked ...
28
votes
1
answer
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Are Minkowski sums of upward closed "convex" sets in $\mathbb{N}^k$ still "convex"? (WAS: Comparing mana costs in Magic: The Gathering)
This was originally a question about comparing mana costs in Magic: The Gathering, but it's turned into a question about Minkowski sums of upward-closed convex sets in $\mathbb{N}^k$. The original ...
28
votes
0
answers
537
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Can every 3-dimensional convex body be trapped in a tetrahedral cage?
Can every 3-dimensional convex body be trapped in a tetrahedral cage?
Although the question is fairly unambiguous, I give all relevant definitions:
$\bullet$ A subset $C$ of $\mathbb{R}^n$ is an $n$-...
26
votes
3
answers
1k
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Largest possible volume of the convex hull of a curve of unit length
What is the largest possible volume of the convex hull of an open/closed curve of unit length in $\mathbb{R}^3$?
25
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4
answers
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Analogy of Parseval identity for Legendre Transform ?
Parseval's identity states that the sum of squares of coefficients of the Fourier transform of a function equals the integral of the square of the function, or
$$ \sum_{-\infty}^{\infty} |c_n|^2 = (1/...
25
votes
4
answers
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Do random projections (approximately) preserve convexity?
The Johnson-Lindenstrauss lemma implies that any set of $k$ points in $\mathbb{R}^d$ can be randomly projected into $d' \approx \log(k)/\epsilon^2$ dimensions such that the distances between each pair ...
25
votes
3
answers
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Research trends in geometry of numbers?
Geometry of numbers was initiated by Hermann Minkowski roughly a hundred years ago. At its heart is the relation between lattices (the group, not the poset) and convex bodies. One of its fundamental ...
25
votes
4
answers
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Ellipse naturally associated with a polygon
My colleagues and I have stumbled onto a way to associate an ellipse, or equivalently a positive definite symmetric matrix, to a polygon that is different from other better known ways. We want to know ...
24
votes
3
answers
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Average measure of intersection of a convex region with its translate
Let $\lambda$ denote the Lebesgue-measure on $\mathbb{R}^n$, and let $C\subset\mathbb{R}^n$ be a convex region.
My question is about
$$f(C):=\int_{C} \lambda(C \cap (x + C) ) \mathrm{d} x.$$
How ...
24
votes
1
answer
1k
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Which natural numbers are a square minus a sum of two squares?
Question: Which natural numbers are of the form $a^2 - b^2 - c^2$ with $a>b+c$?
This question came up in (Eike Hertel, Christian Richter, Tiling Convex Polygons with Congruent Equilateral ...
24
votes
1
answer
521
views
What is the minimal volume of the intersection of a self-dual cone and the unit ball?
When thinking of some other problem, I stumbled upon the following innocently looking question that is natural enough to have been considered (and, possibly, solved) many years ago. However my ...
23
votes
1
answer
647
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Covering the unit sphere in $\mathbf{R}^n$ with $2n$ congruent disks
Let $v_i$ be $2n$ points in $\mathbf{R}^n$, with equal distance $|v_i|$ from the origin. Suppose that the convex hull of these points contains the unit ball. Is it known that $|v_i|\geq\sqrt{n}$? ...
22
votes
5
answers
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Is the diameter of a centrally symmetric convex body realized by a pair of antipodal points?
Let $S \subset \mathbb{R}^n$ be the boundary of a centrally symmetric convex body and provide $S$ with the geodesic metric given by its embedding in Euclidean space (i.e., the distance between two ...
22
votes
2
answers
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Do the elementary properties of mixed volume characterize it uniquely?
Background
Take 2 convex sets in $\mathbb{R}^2$, or 3 convex sets in $\mathbb{R}^3$, or generally, $n$ convex sets in $\mathbb{R}^n$. "Mixed volume" assigns to such a family $A_1, \ldots, A_n$ a ...
21
votes
5
answers
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Is there a midsphere theorem for 4-polytopes?
The (remarkable) midsphere theorem says that each combinatorial
type of convex polyhedron may be realized by one all of whose edges are
tangent to a sphere
(and the realization is unique if the center ...
21
votes
2
answers
1k
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Structures of the space of neural networks
A neural network can be considered as a function
$$\mathbf{R}^m\to\mathbf{R}^n\quad
\text{by}\quad x\mapsto w_N\sigma(h_{N-1}+w_{N-1}\sigma(\dotso h_2+w_2\sigma(h_1+w_1 x)\dotso)),$$
where the $w_i$ ...
21
votes
2
answers
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On convergence of convex bodies
Let $K\subset \mathbb{R}^n$ be a compact convex set of full dimension. Assume that $0\in \partial K$.
Question 1. Is it true that there exists $\varepsilon_0>0$ such that for any $0<\...
20
votes
4
answers
906
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The limit of edge-midpoint convex polyhedra
Starting with a convex polyhedron $P_1 \subset \mathbb{R}^3$,
replace that with $P_2$, the convex hull of the midpoints of the edges of $P_1$.
Continuing this process, we obtain a ...
20
votes
2
answers
871
views
A functional inequality about log-concave functions
Let $f,g$ be smooth even log-concave functions on $\mathbb{R}^{n}$, i.e.,$f=e^{-F(x)}, g=e^{-G(x)}$ for some even convex functions $F(x),G(x)$. Is it true that:
$$
\int_{\mathbb{R}^{n}} \langle \...
19
votes
3
answers
2k
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Cutting convex sets
Any bounded convex set of the Euclidean plane can be cut into two convex pieces of equal area and circumference.
Can one cut every bounded convex set of the Euclidean plane into an arbitrary number $...
19
votes
1
answer
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Where to find some subset of Khovanskii's 15 proofs of the BKK theorem?
(I asked this question on MSE, but someone suggested it would be better asked here.)
I'm a fan of the Bernstein-Khovanskii-Kushnirenko theorem (that the number of solutions in $(\mathbb{C}^*)^n$ to a ...
19
votes
0
answers
621
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Is there a simpler proof of the key lemma in the paper by Hiroshi Iriyeh and Masataka Shibata on the 3D Mahler conjecture?
In this remarkable paper 30 pages are occupied by the proof of the following innocently looking lemma:
Let $K$ be an origin-symmetric convex body in $\mathbb R^3$. There exist three planes through ...
18
votes
2
answers
825
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Reference to a conjecture on unit vectors in Euclidean space
I have heard that there exists the following conjecture (if I am not mistaken).
Let $u_1,\dots,u_n$ be unit vectors in an $n$-dimensional Euclidean vector space. Then there exists another unit vector ...
18
votes
3
answers
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Deciding membership in a convex hull
Given points $u, v_1, \dots,v_n \in \mathbb{R}^m$, decide if $u$ is contained in the convex hull of $v_1, \dots, v_n$.
This can be done efficiently by linear programming (time polynomial in $n,m$) in ...
18
votes
3
answers
918
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Convex functions in convex sets
Suppose $\Omega \subset \mathbb{R}^n$ is some bounded, convex set. For which domains $\Omega$ is it true that for every convex function $f:\Omega \rightarrow \mathbb{R}$ the average of the function in ...
18
votes
3
answers
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Are the Platonic solids shadows of 4-polytopes?
Say that a 3D shadow of a 4-polytope is a parallel projection to 3-space, not necessarily orthogonal to that 3-space (that would make it an orthogonal projection).
I am wondering if each of the five ...
18
votes
1
answer
902
views
Is the ball reducible in some high dimension?
Let $K$ be a bounded symmetric ($-K=K$) open convex body in $\mathbb R^n$. The critical determinant $d(K)$ of $K$ is the least possible volume $|\operatorname{det}(a_1\dots a_n)|$
of the fundamental ...
17
votes
3
answers
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The cone of positive semidefinite matrices is self-dual? (reference needed)
I'm seeking a reference for the following fact.
The cone of positive semidefinite matrices is self-dual (a.k.a. self-polar).
This result is relatively easy to prove, has been known for a long time,...
17
votes
1
answer
519
views
Does the boundary of a convex body contain a regular planar pentagon?
How to prove or disprove that the boundary of any convex body in $\mathbb{R}^3$ includes 5 points which form a regular planar pentagon? The following consideration suggests the answer "yes": if we ...
17
votes
2
answers
978
views
Placing points on a sphere so that no 3 lie close to the same plane
Motivation
I am working with arbitrary parallelopiped tilings given by projection from a higher dimensional space. The collection of tiles, and some properties of the higher dimensional space are ...
17
votes
0
answers
369
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Talagrand's "Creating convexity" conjecture
We say a subset $A$ of $\mathbb{R}^N$ is balanced if
\begin{equation}
x \in A, \lambda \in [-1,1] \implies \lambda x \in A.
\end{equation}
Given a subset $A$ of $\mathbb{R}^N$, we write
\begin{...
17
votes
0
answers
361
views
Is the dodecahedron flexible (as a polytope with fixed edge-lengths)?
Consider the (regular) dodecahedron $D\subset\Bbb R^3$. I want to continuously deform it so that throughout the deformation
it stays a convex polytope,
it stays a combinatorial dodecahedron (i.e. its ...
16
votes
5
answers
1k
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A characterization of convexity
While doing some research on polytopes I came to the following question. Maybe it's already somewhere but anyway I'll post it here.
Let $X\subset \mathbb{R}^3$ be such that, for every plane $P$, $P\...
16
votes
3
answers
902
views
What is known about sufficient conditions for the rigidity of a convex surface?
A convex surface is a connected open subset of the boundary of a convex body in $\mathbb{R}^3$.
An "infinitesimal bending" of a convex surface $S$ is a deformation of $S$ given by a velocity field $v:...
16
votes
1
answer
1k
views
Optimal 8-vertex isoperimetric polyhedron?
I know from Marcel Berger's
Geometry Revealed:
A Jacob's Ladder to Modern Higher Geometry
(p.531)
that it is not yet established which polyhedron in $\mathbb{R}^3$ on 8 vertices achieves the optimal ...