The convex-geometry tag has no wiki summary.

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**1**answer

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### Continuous maps which send intervals of $\mathbb{R}$ to convex subsets of $\mathbb{R}^2$

Let $f : \mathbb{R} \longmapsto \mathbb{R}^2$ 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 line in ...

**36**

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**4**answers

2k views

### 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 noone had ever ...

**30**

votes

**0**answers

949 views

### 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 ...

**26**

votes

**3**answers

2k views

### 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 ...

**24**

votes

**3**answers

696 views

### 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 $\mathbb{R}^3$?

**23**

votes

**1**answer

966 views

### 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 ...

**21**

votes

**4**answers

629 views

### 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 ...

**21**

votes

**4**answers

2k views

### 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 ...

**21**

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**2**answers

1k views

### 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 ...

**19**

votes

**4**answers

639 views

### 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 ...

**18**

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**10**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 ...

**18**

votes

**3**answers

1k views

### 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 ...

**17**

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**4**answers

1k views

### 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 = ...

**17**

votes

**2**answers

803 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 ...

**16**

votes

**3**answers

1k views

### 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 ...

**16**

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**4**answers

770 views

### 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 ...

**16**

votes

**1**answer

311 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 ...

**15**

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**2**answers

867 views

### Intuitive explanation of Dvoretzky's theorem

I am wondering if anyone has an enlightening explanation of why Dvoretzky's theorem (which says that a high-dimensional convex body has an almost round central section) is true -- there are a number ...

**15**

votes

**1**answer

444 views

### Convex bodies with constant maximal section function in odd dimensions

In 1970 or so, Klee asked if a convex body in $\mathbb R^n$ ($n\ge 3$) whose maximal sections by hyperplanes in all directions have the same volume must be a ball. The counterexample in $\mathbb R^4$ ...

**14**

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**4**answers

444 views

### 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 ...

**14**

votes

**2**answers

593 views

### 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 ...

**13**

votes

**2**answers

314 views

### Lattice points and convex bodies

Given are two convex bodies $K, L \subset \mathbb{R}^n$ that contain the origin as an interior point. Assume the number of integer points contained in $\lambda K$ equals the number of integer points ...

**13**

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**2**answers

990 views

### 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 ...

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**0**answers

346 views

### Surface area of convex hull [duplicate]

Let Q be the convex hull of a non-convex polyhedron P. Is it true that the surface area of Q is not greater than the surface area of P?

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**2**answers

224 views

### Smallest containing simplex

Let $V_n$ be the least real number such that for every convex subset of $\mathbb{R}^n$ with hypervolume $1$ there is a containing simplex with hypervolume $V_n$.
What is known about $V_n$? Is there a ...

**12**

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**2**answers

640 views

### Point on a line nearest a point in Banach space

I have a Banach space geometry question (a curiosity-driven spin-off from a research topic). Given a point $x$ on the unit sphere of a Banach space and a vector $y\ne 0$, there is a multiple $t_0y$ of ...

**12**

votes

**1**answer

365 views

### (A question about)${}^3$ 3-dimensional convex bodies

Related to the questions mathoverflow.net question No. 137850 and mathoverflow.net question No. 39127, is there a 3-dimensional convex body other than a ball whose perpendicular projections in all ...

**12**

votes

**1**answer

658 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 ...

**12**

votes

**0**answers

189 views

### How large are the smallest-area projections of a high-dimensional convex body?

Let $B$ be a convex body in $\mathbb{R}^d$, equipped with its standard Euclidean form, and assume that
$$\intop_B x \, dx = 0$$
$$\frac{1}{|B|_d} \intop_B x_i x_j \, dx = \delta_{ij},$$
a ...

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**5**answers

749 views

### 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 $P$, ...

**11**

votes

**3**answers

634 views

### Continuous automorphism groups of normed vector spaces?

Consider the metric space on, say, ℝ2 induced by the various $L^p$ norms, and the group of isometries from that space into itself that preserve the origin. When $p=2$ I get the continuous group ...

**11**

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**3**answers

662 views

### Deciding membership in a convex hull

Problem: 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 ...

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**2**answers

439 views

### Triangle with largest perimeter in a convex region

What is the largest value of $r$ such that the following statement is always true?
"Let $C$ be a convex region with area $1$. There must exist a triangle contained in $C$ whose perimeter is at least ...

**11**

votes

**1**answer

414 views

### Pushing convex bodies together

Given two convex bodies $A$ and $B$, in $\mathbb R^3$ let's say. We define $A(t)$ and $B(t)$ as $A+xt$ and $B+yt$ where $x,y$ are two arbitrary points. (That is the Minkowski sum, so the two bodies ...

**11**

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**3**answers

490 views

### finding the most-isolated point in a high-dimensional cube

I have a set of points {$x_1,\ldots,x_n$} located in the d-dimensional unit cube $[0,1]^d$. $n$ is about 1000 and $d$ is about 25. I'd like to find
$\max_{\omega\in [0,1]^d}\min_{i=1,\ldots,n} ...

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votes

**2**answers

1k views

### 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 ...

**11**

votes

**1**answer

336 views

### Detecting a hidden convex body with line probes

Imagine that, somewhere inside an origin-centered, unit-radius sphere
$S$ in $\mathbb{R}^3$,
sits a convex body $K$ of volume vol$(K)=\alpha (\frac{4}{3} \pi)$,
with $\alpha < 1$ the fraction of ...

**11**

votes

**0**answers

310 views

### Unit ball of smallest volume in a Hilbert geometry

In a letter to Felix Klein published in Mathematische Annalen 1895 (see here), Hilbert generalized the Cayley-Klein model of hyperbolic geometry by defining a metric on the interior of a convex body ...

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**0**answers

742 views

### Constructive aspects of Caratheodory's theorem in convex analysis

Let me paraphrase Caratheodory's theorem in a probabilistic setup:
Let $X$ be a real-valued random variable. For $k = 1, \ldots, m$, let $f_k: \mathbb{R} \to \mathbb{R}$ be a continuous function such ...

**10**

votes

**2**answers

585 views

### Helly theorem + Nerve

Consider nerve $\mathcal N$ of a finite set of convex sets in $\mathbb R^n$.
Helly theorem says that $\mathcal N$ is completely determined by its $n$-skeleton, say $\mathcal N_n$.
It seems that not ...

**10**

votes

**3**answers

965 views

### Isometric (?) embedding problem.

Given a convex surface $S$ in $\mathbb{R}^3,$ you can associate to each point the length of the inward pointing normal contained inside $S.$ Call this quantity $s(x).$ Now, given a positive function ...

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**0**answers

119 views

### Dividing a convex region to minimize average distances

Let $C$ be a convex region in the plane with area 1 that contains distinct points $p_1,\dots,p_n$. Say I'd like to divide $C$ into $n$ pieces $C_1,\dots,C_n$, each of area $1/n$, and I'd like to ...

**9**

votes

**3**answers

1k views

### Area Enclosed by the Convex Hull of a Set of Random Points

Consider $n$ points generated randomly and uniformly on a unit square. What is the expected value of the area (as a function of $n$) enclosed by the convex hull of the set of points?

**9**

votes

**2**answers

239 views

### Can you see smoothness of the boundary of a convex body from its shadow?

Let $d \geq 3$ and suppose that $K \subset \mathbb{R}^{d}$ is a convex body (compact, convex, non-empty interior). Is the following true?
The boundary $\partial K$ is a $C^1$-manifold if and only ...

**9**

votes

**2**answers

536 views

### Concentration of measure for arbitrary convex bodies?

There are various "concentration-of-measure" theorems,
the best known that due to Lévy,
which is this (informally): the volume of a sphere $S^d$ in $d$ dimensions is largely
concentrated around an ...

**9**

votes

**1**answer

776 views

### Reference request: Ehrhart's conjecture on the geometry of numbers

Conjecture (Ehrhart). If a convex body $K \subset {\mathbb R}^n$ has its barycenter at the origin and contains no other point with integer coordinates, the volume of $K$ is less than or equal to $(n ...

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**0**answers

331 views

### How aspherical can a Gömböc be?

A Gömböc is a homogeneous massive convex solid that can rest on a horizontal plane in just two positions of equilibrium under gravity: one stable and the other unstable. How small a proportion of the ...

**8**

votes

**1**answer

240 views

### Largest convex hull of a unit length path

What is the largest area possible for the convex hull of a path of unit length lying on a plane? For what paths is that largest area attained?

**8**

votes

**4**answers

854 views

### Applications of Hilbert's metric

Among the fascinating constructions in mathematics is the Hilbert metric on a bounded convex subset of ${\mathbb R}^n$.
Where, within mathematics, is it used ? I know at least a proof of the ...

**8**

votes

**2**answers

497 views

### Do elongated convex objects all have long simple geodesics?

Let $S$ be a closed convex surface, the boundary of a compact
convex body in $\mathbb{R}^3$.
I am interested in whether there are conditions on its shape
that ensure that it supports a long, simple ...