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3
votes
2answers
78 views

Convex extensibility of combination of two lines

This may be too easy, but: Is there a function $f$ on the first quadrant of $\mathbb R^2$ such that $$ f(x,1)=x,\qquad f(x,0)=0, $$ and $f$ is convex or concave? Note there is no solution of ...
18
votes
2answers
370 views

Trapping a convex body by a finite set of points

In $\mathbb{R}^n$, let $K$ be a convex body and $T$ a finite set of points disjoint from the interior of $K$. Say that $T$ traps $K$ if there is no continuous motion of $K$ carrying $K$ arbitrarily ...
3
votes
1answer
238 views

About generalized Minkowski inequality

For which functions $f:\mathbb{R}^+ \to \mathbb{R}^+$ does the inequality $f^{-1}\left(\sum\limits_{k=1}^n f(x_k+y_k)\right) \leq f^{-1}\left(\sum\limits_{k=1}^n f(x_k)\right) + ...
2
votes
0answers
66 views

Are there pathological examples of log-concave measures that admit no shifts?

Does there exist a random vector $X$ in, say, the space $\mathbb{R}^\infty$ of sequences that has the following properties? The distribution of $X$ is log-concave, i.e. for every $n$ the joint ...
3
votes
1answer
124 views

Is this function of a matrix convex?

Let $\mathcal{N}_{n}$ be the set of symmetric nonnegative irreducible matrices. For a matrix $A \in \mathcal{N}_{n}$ let $v^{A}$ be its Perron vector, normalized so that $||v^{A}||_{2}=1$. Define the ...
1
vote
0answers
54 views

Given a multivariate polynomial with even degree, can we find its tightest convex polynomial 'envelop'?

To be specific, given a multivariate polynomial function $f: \mathbb{R}^n \rightarrow \mathbb{R}$ with even degree $2d$, can we construct a convex polynomial function $g$, such that: $\forall ...
5
votes
2answers
284 views

Can we decompose a polynomial into difference of convex polynomials?

Given a multivariate polynomial $p(x_1, ..., x_n)$ on $\mathbb{R}^n$, can we always decompose it into the difference of two convex polynomials? i.e., is there a pair of convex polynomials $f$ and $g$, ...
-1
votes
1answer
94 views

Exponential Convexity Results [closed]

$\textbf{Definition:}$ 1. A function $h : (a,b)\rightarrow\mathbb{R}$ is exponentially convex if it is continuous and $$\sum _{i, j=1}^n\xi_i\xi_jh(x_i+x_j)\geq 0,$$ for all $n\in\mathbb{N}$ and all ...
3
votes
1answer
232 views

Non-asymptotic large deviations for a convex set

Let $X_1,\dots,X_n$ be $n$ i.i.d random variables taking values in a Polish vector space $\mathcal{X}$ and with (Borel) probability distribution $\mu$. For any convex, compact $\Gamma \subset ...
0
votes
1answer
136 views

Exponential Convexity

$\textbf{Definition:}$ 1. A function $h : (a,b)\rightarrow\mathbb{R}$ is exponentially convex if it is continuous and $$\sum _{i, j=1}^n\xi_i\xi_jh(x_i+x_j)\geq 0,$$ for all $n\in\mathbb{N}$ and all ...
0
votes
0answers
65 views

Open ideally convex sets

Background Recall that a subset $A \subseteq X$ of a Banach$^{1}$ space $X$ is said to be ideally convex if, for every bounded sequence $(x_n)_{n \in {\mathbb N}}$ in $A$ and every sequence ...
11
votes
0answers
180 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 ...
1
vote
1answer
170 views

How to find the necessary and sufficient conditions for a non-convex function to be locally convex?

Let $f(X)\geq 0$ be a nonconvex $C^\infty$ function: $\mathbb R^3\to \mathbb R$. Give any fixed $X_0$ such that $f(X_0)=\epsilon^+$, and the level set: ${L}=\{X\in \mathbb R^3:f(X)\leq \epsilon^+\}$ ...
0
votes
1answer
102 views

Product of derivatives of a convex function [closed]

Given a well-behaved function $f(x,t)$ such that $\frac{\partial f}{\partial x}<0$ and $\frac{\partial^2f}{\partial x^2}>0$, is there a way to show \begin{equation} ...
0
votes
1answer
108 views

Eigenvalues of a given parametrized matrix.

Let $\mathbf{A}$ and $\mathbf{B}$ be two complex rank-one $N\times N$ positive semi-definite matrices. Let the matrix $\mathbf{C}$ be defined as \begin{align} ...
1
vote
0answers
97 views

Proving solution to linear parabolic PDE is convex with negative third derivative

I have a PDE in $g(y,t)$ of the form \begin{equation} a\frac{\partial^2g}{\partial y^2}y^2-b\frac{\partial g}{\partial y}y -rg + \frac{\partial g}{\partial t} - c = 0 \end{equation} in which $a$, $b$, ...
14
votes
0answers
439 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? ...
4
votes
3answers
265 views

Convex hulls of families of probability measures

Let $X$ be a standard Borel space, so that the space of Borel probability measures on $X$ is also a standard Borel space. We denote it by $\mathcal P(X)$. In this paper for any family of probability ...
1
vote
0answers
68 views

Directional derivates and unique subgradients

I have a question about the fine structure of convex functions. Convex functions behave very regular in the interior of their domain of definition (e.g. they are locally Lipschitz continuous there) ...
5
votes
1answer
160 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 $S$. ...
3
votes
1answer
199 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 ...
10
votes
2answers
344 views

Inequalities for averaging over partially ordered sets

Let's start from a classical inequality: If $0\le a_1\le\cdots\le a_k$ and $0\le b_1\le\cdots\le b_k$ then $(a_1+\cdots+a_k)(b_1+\cdots+b_k)\le k(a_1b_1+\cdots+a_k b_k)$. It can be written also in ...
11
votes
1answer
311 views

Tverberg's theorem in CAT(0) spaces

Does Tverberg's theorem hold for CAT(0) spaces of covering dimension $d<\infty$: Is it true that for any $d$-dimensional $CAT(0)$-space $X$ and a subset $E\subset X$ of cardinality $(d + 1)(r - ...
6
votes
0answers
163 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
votes
2answers
290 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 ...
1
vote
0answers
214 views

Proof of an inequality for a linear combination of three trigonometric functions

Given a function $$f(t) = k_{1} \sin(t+\alpha) + k_{3} \sin(3t+\beta) + k_{5} \sin(5t+\gamma)$$ where $k_{1}, k_{3}$ and $k_{5}$ are all positive parameters, and the three phase angles, $ ...
6
votes
0answers
128 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 ...
-1
votes
1answer
47 views

Determining the sign of each element of the optimal of a strict convex function

The problem is: Let $\vec{x}\in\mathbb{R}^d$ be the variable and $f(\vec{x})$ be a scalar function that is globally strictly convex in $\mathbb{R}^d$. We assume the unique optimum of $f$ to be ...
4
votes
1answer
197 views

A construction related to scissors congruence

I was thinking about the following some time ago. My question is whether such things have been studied before. Let $E_n$ be the abelian group with a generator for each (bounded) euclidean polytope of ...
0
votes
1answer
685 views

Is a convex function continuous and almost everywhere differentiable? [closed]

is this statement true ? assume $f:D\rightarrow \mathbf{R} $ is a convex function where $D\subset \mathbf{R}^n$ is a convex set. $f$ is continuous and almost everywhere differentiable and in class ...
1
vote
1answer
198 views

How to examine the convexity of a complex function numerically?

I have a function which does not have a closed form . Large numerical effort should be done to evaluate the function for even a single point. How can I examine the convexity of my function over the ...
0
votes
1answer
183 views

carleman inequality

Is there a connection between carleman inequality discovred by T. Carleman in 1922 if I am not mistaken in his research on quasianalytic functions and what is called Carleman estimates used in the PDE ...
2
votes
0answers
100 views

Why pseudoconvexity is important in Partial differential equation theory?

I am a new researcher in mathematics and I work on convexity. Are convexity and pseudoconvexity related topics and in which respect to PDE theory ? One of the important results in PDE theory is the ...
1
vote
2answers
228 views

Construction of a convex function nondifferentiable on a countable set [closed]

Let $H$ be a countable subset of $[0,1]$. Construct a convex function $f:[0,1]\rightarrow\mathbb{R}$ such that $f$ is nondifferentiable on $H$ and differentiable in the rest.
0
votes
2answers
165 views

Convexity of a minimum function

I was reading a proof of $9g-9$ theorem which states that $9g-9$ length parameters are sufficient the parametrize the Teichmuller space of a closed surface of genus $g$. The proof uses the following ...
4
votes
0answers
114 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 $K_{1}+K_{2}$ ...
1
vote
1answer
132 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 ...
8
votes
1answer
426 views

Small quadrilaterals containing a given convex region

It is easy to prove that (*) Every convex planar set of area 1 is contained in a quadrilateral of area 2. It is also easy to see that statement (*) remains true if the constant 2 is replaced with a ...
1
vote
0answers
58 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 ...
1
vote
0answers
92 views

Linearization of cones

Suppose that $K$ is a closed convex cone in $R^{n}$. Is there a "nice" function $f:R^{n} \rightarrow R^{m}$ so that $f(K)$ is a subspace? What about an approximate subspace?
1
vote
0answers
58 views

Reference request: Compact metrizable semilattices with small connected semilattices are absolute retracts

Let $X$ be a compact metrizable topological semilattice with a neighbourhood base of sub-semilattices that are path connected. I need a reference for a proof that $X$ is an absolute retract. Here is ...
1
vote
1answer
149 views

Helly's number from biconvex functions

Helly's Theorem states the following. Suppose $X_1,X_2,...,X_N$ are convex sets in $\mathbb{R}^d$, such that for any index-set $I$ with $|I| \leq h(d) := d+1$, we have $\bigcap_{i \in I} X_i \neq ...
0
votes
0answers
52 views

convexity of two linear spaces connected by convex nonlinear equality constraints

If there are two sets of linear constraints in different variables, Ax <= b with x_l <= x <= x_u and Cy <= d with y_l <= y <= y_u, and a set of equality constraints of a specific ...
4
votes
1answer
184 views

Zoll Flat Finsler tori and convex bodies on a starry night

The starry night. The "celestial sphere" is given by set of non-zero vectors in $\mathbb{R}^n$ modulo positive dilations (i.e., $v \equiv w$ if $v = \lambda w$ for some $ \lambda > 0$) and the ...
1
vote
2answers
209 views

On a version of gradient descent

I am trying to read this paper and have gotten stuck. The author considers the problem of minimizing a convex function whose gradient has Lipschitz constant $M$ and considers the scheme $$ x(t+1) = ...
1
vote
1answer
77 views

Constrained minimum maximal distance.

Let $C$ and $D$ be two convex sets. And suppose $C\cap D\neq \emptyset$. Let $x^*$ is the solution to the optimization problem: $$\min_{x\in C} \max_{y \in D} |x-y|^2$$ Is it true that $x^* \in D$. ...
0
votes
2answers
241 views

Higher dimensional convex hull

Let $CH(S)$ be a convex hull of a finite set $S$ and denote the set of all the vertices of $CH(S)$ as $Vert(S)$. For a vertex $v \in Vert(S)$, it has an associated set $E(v)$ which is defined as ...
1
vote
1answer
149 views

Regularizing a Convex function with itself

Hi, This is a problem that has being bothering me the last few days. Assume a convex function $f(x): {\mathbb R}^n \rightarrow {\mathbb R}$ with a unique minimizer $x^{\star}$. Now consider the ...
2
votes
3answers
238 views

Measuring the distance of a convex set from a ball (Nikodym distance)

Assume we have a centrally symmetric convex set $K \subset \mathbb{R}^n$ such that Vol(K)=1. In addition, assume that for every direction $u$ we know that $Vol(K \Delta R_u(K)) < \epsilon$, where ...
1
vote
0answers
60 views

Which matrix/operator in a cone has the largest negative spectral part?

Background: Let $\mathcal{K}$ be set (convex cone, if you like) of symmetric matrices of order $n$. Each matrix $A \in \mathcal{K}$ can be decomposed in a unique way as $A=A_{+}-A_{-}$, where ...