Questions tagged [convexity]

For questions involving the concept of convexity

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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 ...
Mohammad Ghomi's user avatar
36 votes
0 answers
1k 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 ...
Anton Petrunin's user avatar
17 votes
0 answers
370 views

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{...
Samuel Johnston's user avatar
16 votes
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2k views

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? ...
alvarezpaiva's user avatar
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14 votes
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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 ...
Alexander Shamov's user avatar
11 votes
0 answers
356 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 ...
Brian's user avatar
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10 votes
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215 views

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 ...
Lviv Scottish Book's user avatar
10 votes
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143 views

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}^...
Denis Serre's user avatar
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10 votes
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253 views

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 ...
Mohammad Ghomi's user avatar
10 votes
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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 ...
Menachem's user avatar
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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 ...
Kurt Luoto's user avatar
8 votes
0 answers
185 views

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^...
Wolfgang's user avatar
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8 votes
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192 views

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)=\...
user_lambda's user avatar
8 votes
0 answers
814 views

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 ...
alvarezpaiva's user avatar
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7 votes
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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:...
Piotr Hajlasz's user avatar
7 votes
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109 views

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\...
alesia's user avatar
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869 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 ...
asv's user avatar
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7 votes
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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, $f^{-1}:[0,\infty)\to[0,\infty)$...
Daniel Spector's user avatar
6 votes
0 answers
132 views

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 $X'$ ...
Anton Petrunin's user avatar
6 votes
0 answers
113 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 $f_k \to f$ ...
Hugo's user avatar
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251 views

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 ...
user_lambda's user avatar
6 votes
0 answers
187 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 ...
alvarezpaiva's user avatar
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6 votes
0 answers
379 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 ...
Yoav Kallus's user avatar
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6 votes
1 answer
285 views

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 ...
asv's user avatar
  • 21.1k
5 votes
2 answers
170 views

Convex hull of bivariate normal random points

Let $n > 1$, and $p_1, \ldots, p_n \in \mathbb{R}^2$ be iid random points following a bivariate normal distribution (doesn't matter which one), and let $X_n$ be the number of vertices of convex ...
Mikhail Tikhomirov's user avatar
5 votes
0 answers
101 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 ...
guest61's user avatar
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0 answers
208 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 ...
Johnny T.'s user avatar
  • 3,547
5 votes
0 answers
239 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 ...
Rafael's user avatar
  • 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: $$\sigma_1(X)=X_1+\cdots+X_n,\quad\ldots\quad,\sigma_n(X)=X_1\cdots X_n.$$ ...
Denis Serre's user avatar
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5 votes
0 answers
460 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 ...
David Eppstein's user avatar
5 votes
0 answers
150 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,...
Hans's user avatar
  • 2,863
5 votes
0 answers
515 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 ...
Johnny Yin's user avatar
5 votes
0 answers
150 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 ...
Andy Sanders's user avatar
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4 votes
0 answers
130 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{...
Motaka's user avatar
  • 291
4 votes
0 answers
144 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 ...
Oliver Roche-Newton's user avatar
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):=(\...
asv's user avatar
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4 votes
0 answers
63 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 ...
Taras Banakh's user avatar
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4 votes
0 answers
345 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(\...
James Silipo's user avatar
4 votes
0 answers
68 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 ...
Denis Serre's user avatar
  • 51.5k
4 votes
0 answers
534 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 \{...
D.E.G.Z.'s user avatar
4 votes
0 answers
235 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}|...
Cristóbal Guzmán's user avatar
4 votes
0 answers
322 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 ...
Joonas Ilmavirta's user avatar
4 votes
0 answers
182 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\...
Richard Senn's user avatar
4 votes
0 answers
149 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) ( \...
Changwang Zhang's user avatar
4 votes
0 answers
292 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 ...
Evan DeCorte's user avatar
4 votes
0 answers
158 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 $K_{1}+K_{2}$ ...
Felix Goldberg's user avatar
4 votes
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 ...
Ian Calvert's user avatar
3 votes
0 answers
112 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 ...
Tom Leinster's user avatar
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3 votes
0 answers
492 views

Prove concavity of real valued function on the non-negative real axis

Fix $\alpha >0$ and define $f_{\alpha}(x) := \ln(\Phi(\alpha-x)-\Phi(-\alpha-x))$, where $\Phi(x)$ is the normal cumulative density function. For some research, I am trying to verify that the ...
Barreto's user avatar
  • 139
3 votes
0 answers
54 views

Injectivity of a convex combination of squares $A^*A$ in $\ell^\infty$

Consider two operators $A,B: \ell^p \to \ell^p$ ([defined and]bounded for all $p \in [1,\infty]$) as well as their adjoints $A^*,B^*:\ell^p \to \ell^p$. Assume $A^*A$ and $B^*B$ have trivial kernel ...
ARG's user avatar
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