Questions tagged [convexity]
For questions involving the concept of convexity
189 questions with no upvoted or accepted answers
38
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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 ...
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36
votes
0
answers
1k
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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 ...
- 45k
19
votes
0
answers
552
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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{...
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16
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0
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2k
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An open problem in convex geometry
Is it possible to find four norms $\| \cdot\|_k$ $( 1 \leq k \leq 4)$ on the plane such that a three-dimensional normed space containing four subspaces isometric to these normed planes does not exist? ...
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14
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0
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310
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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 ...
- 4,010
11
votes
0
answers
364
views
Is there yet an example of a non-negative convex polynomial that cannot be written as a sum-of-squares?
I have read that it remains an open question, whether an example can be constructed of a non-negative convex polynomial that cannot be written as a sum-of-squares. My reading includes the following ...
- 173
10
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0
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226
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Extremal bases in finite-dimensional Banach spaces
Definition. A basis $e_1,\dots,e_n$ for a Banach space $X$ is called extremal if there exists a point $s$ in the unit sphere $S_X=\{x\in X:\|x\|=1\}$ such that for every $i\in\{1,\dots,n\}$ the ...
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10
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0
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143
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A geometrical problem in terms of a convex function
I wish to know whether the following problem has ever been investigated.
Let $D$ be a convex domain in ${\mathbb R}^d$, with smooth boundary $\partial D$. Let $\vec V:\partial D\rightarrow{\mathbb R}^...
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10
votes
0
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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
10
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418
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Determining convexity of a polygon from its Fourier coefficients
Consider an $n$-sided polygonal curve in the plane, represented by an ordered set of points $(x_0, x_1, \ldots, x_{n-1})$; line segments connect consecutive points and also $x_{n-1}$ to $x_0$. It is ...
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10
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0
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302
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Are plactic classes convex under the right weak Bruhat order?
For those who are unfamiliar with the terminology, I'll explain a little.
The symmetric group $S_n$, as a type A Coxeter group, has generators $\{s_1,\ldots,s_{n-1}\}$ with relations (1) $s_i^2$ for ...
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8
votes
0
answers
460
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Are there any characterizations of $C^2$ convex functions?
There are several characterizations of convex functions with the Lipschitz continuous gradient. If we already know that the function is of class $C^1$, then we have the following equivalent conditions:...
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8
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0
answers
194
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Geometric mean of three or more positive definite matrices
The geometric mean of two positive definite (Hermitian) matrices of same size is defined by $$A\natural B := A^{1/2}(A^{-1/2}BA^{-1/2})^{1/2}A^{1/2},$$equivalently,
$$A\natural B =(BA^{-1})^{1/2}A=A(A^...
- 13.4k
8
votes
0
answers
210
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Concavity of product and ratio of sums
Apologies if this question is not appropriate for MathOverflow. I have asked at Math.StackExchange without success.
Consider the function $f:\mathbb{R}^n\rightarrow \mathbb{R}$ defined as
$$
f(x)=\...
- 379
8
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0
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Geometry of the metric cone
Let us say that two metrics $d$ and $d_0$ on a set $X$ are related if there exist positive constants $0 < \alpha \leq \beta$ such that
$$
\alpha \,\left(d_0(x,y) + d_0(y,z) - d_0(x,z)\right) \leq
...
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7
votes
0
answers
120
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What is the closed cone generated by constant and coordinate functions and closed under taking $f\mapsto\max(f,0)$?
Let $C$ be the smallest closed convex cone of functions from $\mathbb{R}^n$ to $\mathbb{R}$ that contains all constant functions, all coordinate functions, and such that $\max(f,0)\in C$ whenever $f\...
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7
votes
0
answers
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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 ...
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7
votes
0
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708
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Minkowski's Inequality for Integrals in Orlicz spaces
EDIT: I have changed the question to have less parameters, fitting it into the context of Orlicz spaces.
Suppose $f:[0,\infty)\to[0,\infty)$ is convex and increasing, $f^{-1}:[0,\infty)\to[0,\infty)$...
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6
votes
0
answers
48
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Strengthening the Kovner-Besicovich theorem: Does every unit-area convex set in the plane contain a centrally symmetric hexagon of area $2/3$?
The Kovner-Besicovich theorem states that every convex set $S$ in the plane contains a centrally symmetric subset $C$ of at least $2/3$ the area of $S$, and that this bound is sharp for triangular $S$....
- 3,068
6
votes
0
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134
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Nearby convex set in a nearby space
Let $K$ be a convex set in a CAT(0) space $X$. Suppose $X'$ is a CAT(0) space that is very close to $X$.
Is there a convex set $K'\subset X'$ that is close to $K\subset X$?
Two spaces $X$ and $X'$ ...
- 45k
6
votes
0
answers
115
views
Positive splitting of Sobolev convergence
Let $f,g,h \in H^1(\mathbb{R}^n)$ be non-negative Sobolev functions such that $f^2 = g^2 + h^2$. Let also $\{f_k\} \subseteq H^1(\mathbb{R}^n)$ be non-negative Sobolev functions such that $f_k \to f$ ...
- 201
6
votes
0
answers
255
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Concavity of a function implicitly defined by a polynomial
Consider the following system of $n$ equations:
\begin{equation}f_j^2 = x_j^2\sum_{i=1}^n A_{ij} f_i
\tag{$\star$}
\end{equation}
where $A_{ij}\geq 0$ are known constants and where $x_j>0$ for ...
- 379
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
6
votes
0
answers
387
views
Local minimum from directional derivatives in the space of convex bodies
I have a function $f(K)$ defined on the space of three-dimensional convex bodies for which I want to show that the unit ball $B$ is a local minimum. I have been able to show if $K$ is not homothetic ...
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6
votes
1
answer
295
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A conjecture (or theorem?) on unit vectors in a Euclidean space
I have heard (if I am not mistaken) that there exists the following conjecture (or theorem?).
Let $u_1,\dots,u_n$ be unit vectors in an $n$-dimensional Euclidean vector space. Then there exists ...
- 21.8k
5
votes
0
answers
106
views
Semilinear elliptic equation
Assume $u$ is a smooth solution for
$$
\Delta u + f(u)=0\qquad \hbox{in}\quad \Omega
$$
and $\Omega$ is a smooth convex domain in $\mathbb{R}^n$.
Is there a conjecture which are the weakest conditions ...
- 329
5
votes
0
answers
233
views
Reference request for convex geometry?
I am looking for a reference for an elementary convex geometry.
In Appendix A (page 1810) of this paper by Green and Tao, they cover some basic results from elementary convex geometry. The results ...
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5
votes
0
answers
275
views
Log-concavity of lattice-functions and convolution
I was looking at the definition of log-concavity:
A function $F:\mathbb{R}^n\rightarrow\mathbb{R}$ is said log-concave
iff $F(x)\geq 0\forall x\in\mathbb{R}^n$ and
$$F(x)^\lambda F(y)^{1-\lambda}\leq ...
- 93
5
votes
0
answers
60
views
Self-duality of cones associated with elementary symmetric polynomials
Let $n\ge3$ be an integer, and denote $\sigma_1,\ldots,\sigma_n$ the elementary symmetric polynomials in $n$ indeterminates:
$$\sigma_1(X)=X_1+\cdots+X_n,\quad\ldots\quad,\sigma_n(X)=X_1\cdots X_n.$$
...
- 52.3k
5
votes
0
answers
481
views
Open convex hull of a closed set
Let $X$ be a closed set in a Euclidean space of finite dimension and suppose that its convex hull $H$ is open. I can prove that, in this case, $H$ is a Cartesian product of a line with an open convex ...
- 18.6k
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
5
votes
0
answers
548
views
Log-concave polynomial is a log-concave function?
A polynomial $\sum\limits_{k=0}^n a_kx^k$ is log-concave if $a_0,\ldots,a_n$ constitute a log-concave sequence. I wonder whether the log-concave polynomial is also a log-concave function with respect ...
- 79
5
votes
0
answers
152
views
Star shaped sets with a midpoint
Suppose $U$ is an open subset of $\mathbb{R}^n$ which is star shaped with respect to $p\in U$. I'll call $p$ a midpoint of $U$ if for any line $\ell$ through $p$, the point $p$ is the midpoint of the ...
- 3,020
4
votes
0
answers
132
views
A Lipschitzian's condition for the measure of nonconvexity
I'm actually working on the measure of nonconvexity and its application. Especially, the Eisenfeld–Lakshmikantham MNC defined - in a Banach space - by:
$$\alpha(A)=\sup_{b\in\overline{\operatorname{...
- 291
4
votes
0
answers
154
views
How many convex or concave subsets are contained in an arbitrary set of $n$ real numbers?
This question is closely related to this post. A set $A=\{a_1<a_2<\dots<a_n\} \subset \mathbb R$ is said to be convex if the consecutive differences are non-decreasing, i.e. if $a_{j+1} - a_j ...
- 1,498
4
votes
0
answers
63
views
Length of curves on convex hypersurfaces
Let $\gamma\colon[a,b] \to \mathbb{R}^n$ be a smooth curve.
Let $f_i\to f$ be a sequence of convex functions on $\mathbb{R}^n$ converging uniformly on compact subsets to $f$.
Let $\hat\gamma(t):=(\...
- 21.8k
4
votes
0
answers
65
views
A standard name of a strongly extremal point of a convex set
I need to name somehow points $x$ of a bounded convex set $C$ in a Banach space $X$ such that the set $$\{x^*\in X^*:x^*(x)=\max x^*[C]\}$$ of support functionals at $x$ has non-empty interior in the ...
- 41.8k
4
votes
0
answers
367
views
On intrinsic volumes
Let $\Gamma$ be a convex polytope in $\mathbb R^n$. The $k$-th intrinsic volume of $\Gamma$ is the number
$$
\text{v}_k(\Gamma)=\sum_{\Delta\in{\mathcal B}(\Gamma,k)}\text{vol}_k(\Delta)\psi_\Gamma(\...
- 311
4
votes
0
answers
70
views
"Singularly convex" cones of matrices
The ambient space if ${\bf M}_n({\mathbb R})$.
Let us begin with facts.
1- The cone of positive semi-definite symmetric matrices is convex. 2- It is a little subtler that the cone $K^+$ of matrices ...
- 52.3k
4
votes
0
answers
535
views
Convexity of the electrostatic energy on a Riemann surface
Let $M$ be a compact Riemann surface.
Let $\Lambda$ be a differentiable real $2$-form of integral one.
Let $G$ be the Green function associated to $\Lambda$, i.e.
$G: M \times M \to \mathbb R \cup \{...
- 41
4
votes
0
answers
244
views
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
$$ \|X\|_{p_1,p_2} = \left( \sum_{j=1}^n \left( \sum_{i=1}^m |x_{ij}|...
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votes
0
answers
324
views
How to check if a manifold can be foliated by strictly convex hypersurfaces?
Let $M$ be a compact Riemannian manifold with boundary.
How can one recognize whether the manifold can be foliated by strictly convex hypersurfaces?
An exact definition is given below.
If the ...
- 8,086
4
votes
0
answers
184
views
This function looks quasiconvex, can't understand why
Suppose that $\mathbf{C}$ is a given matrix with non-negative entries in $\mathbb{R}^{m\times n}$ and $d$ is a given scalar, and let $g(\mathbf{y})$ be defined by $$g(\mathbf{y}):=\max_{\mathbf{x}\in\...
- 41
4
votes
0
answers
151
views
Prove a complicated function (in epidemic spreading search) to be convex
When analyse epidemic spreading, I came across to prove that a complicated function $f(x)$ is convex when $0 \leq x \leq 1$.
\begin{equation}
f(x)=\frac{b_1g'(x) f_1(x)^{n-2}+g'(1) \gamma}{g'(1) ( \...
- 185
4
votes
0
answers
293
views
When is the sum of a weak-$*$ closed convex cone and a subspace also weak-$*$ closed?
Let $X$ be a Banach space. Suppose $C \subset X^*$ is a convex cone and $V \subset X^*$ is a subspace, and suppose both $C$ and $V$ are closed in the weak-$*$ topology. Are there any general ...
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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
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 $K_{1}+K_{2}$ ...
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4
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0
answers
119
views
Symmetry of Sundry Planar Convex Sets of Constant Width & Minimal Area
In a much broader paper in “Optimization Methods & Software 27,6 (2012) pp1073-1099” Bayen & Henrion consider planar compact, convex sets with support functions which are finite Fourier ...
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3
votes
0
answers
77
views
Is the norm of first or second level of of signature a convex function?
I understand this is not a research level question but I really want to know, would anyone please help.
This question is related to the signatures that arises in rough path theory. https://en....
- 495
3
votes
0
answers
208
views
Reference request: Carathéodory-type theorem for convex hulls of closed sets
I'm looking for a reference for the following theorem.
Theorem Let $X$ be a closed subset of $\mathbb{R}^N$, and let $a$ be a point of its convex hull $\operatorname{conv}(X)$. Then there exist ...
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