All Questions
Tagged with convexity convex-geometry
131 questions
0
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0
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56
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linear functions/hyperplanes vs. convex functions/convex sets in Hilbert space
The simplest Hahn-Banach extension theorem in Hilbert space $X$ avoids the use of the axiom of choice by virtue of the Riesz representation theorem. But what about the version of the theorem where the ...
- 1,461
4
votes
1
answer
2k
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Is an ambiguity set with Wasserstein distance of order 1 is convex?
I have a question about the convexity of an Wasserstein ambiguity set.
Let $W_1(\mu, \nu)$ be the Wasserstein distance of order 1 between $\mu$ and $\nu$, defined as
$$W_1(\mu, \nu) := \min\limits_{\...
- 51
3
votes
1
answer
258
views
Polygon of convex arcs
Convex polygons in the plane $R^2$ arise in linear programming where the constraints are linear. The objective linear function attains its maximum at a vertex of the feasible region(if exists).
Assume ...
- 422
0
votes
3
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111
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On the properness of the graph of a convex function
Let $f : \Omega \subseteq \mathbb{R}^n \to \mathbb{R}$ be a smooth and convex function. Let us assume that $\Gamma_f = \mathrm{graph}(f) $ is a complete hypersurface of $\mathbb{R}^{n+1}$. Then I know ...
- 823
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0
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69
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Choquet Theorem for the cone of non-negative operators
Let $\mathcal B_+$ be the convex cone of bounded non-negative self-adjoint operators on $L^2(\mathbb R)$. With the perspective of applying Choquet's Theorem, I would like to know if the extreme points ...
- 16.2k
5
votes
1
answer
161
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Cutting a convex body into two congruent pieces
This question is related to How to make a sandwich from just one piece of bread?, asked on Feb 23 '17 by erz, and it goes as follows:
Question. If a convex closed and bounded region $C$ in the ...
- 7,276
3
votes
2
answers
371
views
Extreme points of a convex set
Let $S$ denote the set of all complex non-negative definite matrices with all diagonal elements being less that or equal to one. Can we show that any matrix which belongs to the set of all non-zero ...
- 455
11
votes
2
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559
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Convex hull of the Stiefel manifold with non-negativity constraints
Consider the Stiefel manifold
$$\mathrm{St}(n,k) :=\{X \in \mathbb{R}^{n\times k} : X^TX = I_k\},$$
where $I_k$ is the $k$-dimensional identity matrix. It is well known that
$$\mathrm{conv} \left( ...
- 1,991
5
votes
0
answers
151
views
Dimensions of faces of convex hull of convex bodies
Let $K_1,\ldots,K_m\subset\mathbb{R}^n$ with $m\geq n$ some convex bodies (i.e. compact with nonempty interior). I am interested in sufficient criteria for the convex hull $K=\textrm{conv}(K_1,\ldots,...
- 3,031
2
votes
1
answer
76
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Common boundary point of convex bodies
Let $K_1, \ldots, K_n \subset \mathbb{R}^n$ be some convex bodies (i.e., compact with nonempty interior) such that for each subset $I \subset \{1,\ldots,n\}$ the set
$$\left( \bigcap_{i \in I} K_i \...
- 3,031
8
votes
1
answer
276
views
Almost convex combinations in $\mathbb R^n$
Working on some problems in the $C_p$-theory I discovered the following simple but amazing
Fact. For any subset $A\subset \mathbb R^n$, non-zero vector $a\in \bar A\subset\mathbb R^n$ and $\...
- 41.8k
9
votes
3
answers
790
views
Faces of the intersection of convex sets
Let $V$ be a normed real vector space and let $K_1, K_2\subseteq V$ be closed convex subsets such that the intersection $K_1\cap K_2$ is non-empty. Assume that $F_1$ is a face of $K_1$ and $F_2$ is a ...
- 391
10
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265
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Plank invariant measures on convex bodies
Let $K\subset R^2$ be a convex body, i.e., a compact convex set with interior points. A plank $P$ is the region between a pair of parallel lines in $R^2$. Let us say that $P$ intersects $K$ properly ...
- 7,149
38
votes
0
answers
1k
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Converse of the Archimedean property of the sphere
In his remarkable book On the Sphere and Cylinder, where he came tantalizingly close to discovering calculus, Archimedes showed that the area of the portion of the sphere contained between a pair of ...
- 7,149
24
votes
3
answers
1k
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Average measure of intersection of a convex region with its translate
Let $\lambda$ denote the Lebesgue-measure on $\mathbb{R}^n$, and let $C\subset\mathbb{R}^n$ be a convex region.
My question is about
$$f(C):=\int_{C} \lambda(C \cap (x + C) ) \mathrm{d} x.$$
How ...
- 343
7
votes
1
answer
145
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Monotonicity of canonical ellipsoids
Let $\mathcal{C}$ be the set of compact convex centrally symmetric sets in $\mathbb{R}^d$, and let $\mathcal{E} \subset \mathcal{C}$ be the set of ellipsoids centered at the origin.
I'm looking for a ...
- 2,479
7
votes
0
answers
904
views
Geometry of level sets of a convex function
EDIT: Let $\Omega\subset \mathbb{R}^n$ be an open subset. Let $f\colon \Omega\to\mathbb{R}$ be a function such that for some $\lambda$ the function $f(x)+\lambda |x|^2$ is convex. Assume that the ...
- 21.8k
3
votes
1
answer
449
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VC dimension under projection
Let $C$ be a family of convex sets in $\mathbb{R}^d$ and assume further that $C$ is closed under translation: for all $A\in C$ and $x\in\mathbb{R}^d$, we have $A+x\in C$.
Let $P:\mathbb{R}^d\to\...
- 6,541
5
votes
1
answer
671
views
When minimum of two supporting functionals of convex bodies is convex?
For a convex compact set $K\subset \mathbb{R}^n$ let us denote by $h_K$ its supporting functional
$$h_K(\xi):=\sup_{x\in K}\langle\xi,x\rangle.$$
Thus $h_K\colon \mathbb{R}^n\to \mathbb{R}$ is a ...
- 21.8k
5
votes
2
answers
320
views
Convex hull with genus information
Are there convexity generalizations that admit genus information?
For example in genus $1$ is there a way to think of this polyhedron as convex while this polyhedron as non-convex? Any two points can ...
- 13.9k
39
votes
2
answers
2k
views
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 ...
- 5,529
4
votes
1
answer
4k
views
weak*-closed convex = closed convex? [closed]
If $X'$ is the topological dual of a Banach space, then is that true that a convex set is closed (for the norm on $X'$ given by $\lVert f \rVert_{X'} := \sup \frac{\langle f , x \rangle}{\lVert x \...
- 2,306
0
votes
0
answers
243
views
Limit of argmin of sum
Suppose that I know $f_n\rightarrow f$ and $g_n\rightarrow g$ are both continuous maps from a Complete Riemmanian Manifold $X$ to $\mathbb{R}$ which converge pointwise almost everywhere. Then is it ...
- 5,405
8
votes
3
answers
337
views
Shape whose translated and scaled copies are closed under intersection
The translated and scaled copies of an equilateral triangle with fixed orientation are closed under intersection - the intersection is again an equilateral triangle with the same orientation.
What ...
- 183
1
vote
0
answers
127
views
Is support function of a convex curve in $\mathbb{R}^2$ absolutely continuous? [closed]
There is an example of a convex function of two variables that is convex but not absolutely continuous. The level set of this function is a convex curve.
To construct one example of such a function ...
- 11
3
votes
1
answer
151
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The center of a minimal convex superbody
Is the following true?
CONJECTURE: $\,$ Let $\ B\ C\subseteq\mathbb R^n\ $ be convex bodies in $\mathbb R^n$ such that $\ C\ $ is centrally symmetric, $\ B\subseteq C,\ $ and $\ t\!\cdot\! B\ $ cannot ...
- 7,500
21
votes
2
answers
1k
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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<\...
- 21.8k
5
votes
1
answer
498
views
Questions about the regularity of the "norm" associated to a convex set
Suppose $K\subset \mathbb{R} ^n$ is a closed convex set whose interior contains the origin. We can assign a gauge function to $K$ as $g_{K}(x):=\inf\{\lambda>0 \mid x\in\lambda K\}$. $g_K$ has all ...
- 755
20
votes
2
answers
922
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 \...
- 3,946
1
vote
1
answer
373
views
does every compact convex set in c0 have but countably many extreme points
This seems plausible, given the properties of the unit ball of $c_0$.
I have a compact set in a complex Banach space $X$ whose closed convex hull has uncountably many extreme points. It would be ...
- 1,591
6
votes
1
answer
309
views
Linear projections of convex sets with unique preimages of boundary points
Fix a compact convex subset $C \subset \mathbb{R}^n$ with nonempty interior. For any subspace $S \subset \mathbb{R}^n$, let $P_S$ denote the orthogonal linear projection onto $S$. I'd like to claim ...
- 630
4
votes
0
answers
244
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On the modulus of convexity of mixed-norm $\ell_{p_1,p_2}$ spaces
Let $\ell_{p_1,p_2}=(\mathbb{R}^{m\times n},\|\cdot\|_{p_1,p_2})$ be the space of $m\times n$ matrices endowed with the mixed-norm
$$ \|X\|_{p_1,p_2} = \left( \sum_{j=1}^n \left( \sum_{i=1}^m |x_{ij}|...
- 980
68
votes
2
answers
2k
views
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 ...
- 629
0
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0
answers
62
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Looking for a homogeneous function with some properties
I'm looking for a 1-homogeneous function $\pi \colon \mathbb{R}^n_{\geq 0} \to \mathbb{R}$ satisfying the following properties:
1) $\pi$ is not concave. This is equivalent to the fact that there ...
- 357
0
votes
0
answers
73
views
Integral representation formula for convex
For $u \in \mathbb{S}^{d-1} \subset \mathbb{R}^d$, it is easy to show that:
\begin{equation}
u=c_d \int_{\mathbb{S^{d-1}}} \xi \mathbb{1}_{\left\{x \cdot u >0 \right\}}(\xi) \ \rm{d}\sigma_{d-1}(\...
7
votes
1
answer
444
views
Is an infinite-dimensional "Lebesgue measure" uniquely determined by a set of positive finite measure?
Let $\mu$ be a probability measure on a subset $C \subset \mathbb{R}^\infty$ of the space of sequences, and assume, for simplicity, that $C$ is closed and convex.
We say that $\mu$ admits shifts if ...
- 4,000
2
votes
0
answers
106
views
Regularity of the Minkowski functionnal of a convex
Let $K$ be a convex compact set in $\mathbb{R}^2$ with $0 \in \overset{\circ}{K}$. The Minkowski functional associated to K is:
\begin{align*}
\varphi_K(x):= \inf \left\{t>0 \; : \; tx \in K \...
- 21
4
votes
2
answers
981
views
Convex Sets and Nearest Neighbors
For a set $S \subseteq \mathbb{R}^n$ and a point $x \in \mathbb{R}^n$, let $c_S(x)$ be the point $s \in S$ that minimizes $\|x-s\|$ if such a point exists and is unique. It is known that $c(x) = s$ ...
- 177
2
votes
1
answer
171
views
Distance from constant width bodies
EDIT As @David has observed, my conjecture was clearly wrong for $\ n:=2.\ $ Let me still give it a chance for $\ n\ge 3$.
I'll call a family $\ F\ $ of bound closed convex subsets of $\ \mathbb R^n\ ...
- 7,500
6
votes
2
answers
1k
views
When does a cone contain its dual cone?
Let $V$ be a finite-dimensional vector space with an inner-product $(,)$ and let $C\subset V$ be a cone in $V$. Let $C^\vee$ denote the dual of $C$ with respect to $(,)$, i.e., the set of vector $v\in ...
- 587
3
votes
0
answers
141
views
Covering fat objects with fat objects
The family of rectangles has the cover property, i.e.:
For every $R\geq 1$, $k\geq 1$: every rectangle with aspect ratio $kR$ can be exactly covered by $\lceil k\rceil$ (possibly overlapping) ...
- 3,549
4
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0
answers
350
views
Convex functions with non-singular hessian measure are continuously differentiable?
It is known that every convex function $f: \Omega\to \mathbb{R}$, $\Omega$ convex subset of $\mathbb{R}^n$, has a weak derivative of bounded variation $Df\in BV_{loc}(\mathbb{R}^n)$ (e.g. Evans and ...
- 1,371
14
votes
0
answers
310
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 ...
- 4,000
7
votes
1
answer
412
views
Shortest curve with given convex hull
Suppose $S\subset\mathbb{R}^2$ is compact and convex. Suppose $\Gamma:[0,1]\to S$ is a continuous curve that passes through every extreme point of $S$, i.e., the convex hull of $\Gamma([0,1])$ is $S$. ...
- 363
3
votes
1
answer
247
views
Map from a convex polygon that increases distance
At the risk of asking an extremely stupid question, suppose that $P\subset\mathbb{R}^2$ is a convex polygon with area $1$ that contains the origin, and let $r$ denote the farthest distance between the ...
- 33
13
votes
1
answer
442
views
Choquet theory and Hilbert's fourth problem
The following text is an attempt to see Hilbert's fourth problem in a new light.
Definition. A pseudometric $d$ on $\mathbb{R}^n$ is called projective if whenever a point $z$ belongs to a line ...
- 13.5k
14
votes
2
answers
882
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 ...
- 13.5k
6
votes
0
answers
189
views
Variations on a problem of S. Mazur
In problem 76 of the Scottish Book Mazur asked
Given a convex body $K$ in three-dimensional space and a point $o$ in its interior, consider the surface $S$ formed by all points $p$ such that the ...
- 13.5k
4
votes
0
answers
162
views
Hilbert metric of a sum of cones
Suppose that $K_{1}$ and $K_{2}$ are pointed closed cones in a finite-dimensional space $V$ whose Hilbert metrics $d_{1},d_{2}$ are known. Is there a way to express the Hilbert metric of $K_{1}+K_{2}$ ...
- 6,962
1
vote
1
answer
163
views
A question concerning convex functions
Consider points $a,b,c$ (not on a line) and $x_1,...,x_n$ in $\Bbb{R}^2$. I am looking for a necessary and sufficient condition in terms of the geometric configuration of these points such that for ...
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