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**5**

votes

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159 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
...

**5**

votes

**2**answers

158 views

### Smooth convex extensibility of combination of two line segments

This is a refined version of my earlier question Convex extensibility of combination of two lines.
Is there a smooth function $f:[0,1]\times [0,1]\rightarrow\mathbb R$ such
that for all $x\in ...

**17**

votes

**2**answers

752 views

### An equivalence relation for norms

Let us say that two norms $\|\cdot\|_1$ and $\|\cdot\|_2$ on a real vector space $V$ are strongly equivalent if there exists a constant $\lambda \geq 1$ such that
$$
\frac{1}{\lambda} \left( \|x\|_1 ...

**4**

votes

**1**answer

132 views

### the validity of a basic statement involving the Hausdorff distance

Let $\Omega_1 \supset \Omega_2 \supset \cdots$ be a sequence of nonempty, open, bounded and convex sets in $R^n.$
Define $\Omega = \operatorname{int} \Bigl( \overline{\bigcap_{k=1}^{\infty} \Omega_k ...

**3**

votes

**2**answers

85 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 ...

**19**

votes

**2**answers

397 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 ...

**4**

votes

**1**answer

424 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) + ...

**3**

votes

**0**answers

78 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

**1**answer

136 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 ...

**2**

votes

**0**answers

170 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

**2**answers

361 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 $g$, ...

**0**

votes

**1**answer

110 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 ...

**5**

votes

**2**answers

432 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 ...

**1**

vote

**1**answer

147 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 ...

**12**

votes

**0**answers

208 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

**1**answer

361 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

**1**answer

115 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

**1**answer

131 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

**0**answers

146 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

**0**answers

1k 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? ...

**5**

votes

**3**answers

483 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

**0**answers

87 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) ...

**6**

votes

**1**answer

223 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

**1**answer

214 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

**2**answers

393 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

**1**answer

363 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 - ...

**11**

votes

**1**answer

299 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

**2**answers

409 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

**0**answers

265 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

**0**answers

142 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

**1**answer

55 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

**1**answer

227 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 ...

**1**

vote

**1**answer

1k 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 ...

**2**

votes

**1**answer

302 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

**1**answer

205 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

**0**answers

113 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

**2**answers

375 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

**2**answers

922 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

**0**answers

126 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

**1**answer

146 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 ...

**9**

votes

**1**answer

485 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 ...

**4**

votes

**0**answers

106 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

**0**answers

112 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

**0**answers

60 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

**1**answer

156 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

**0**answers

58 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

**1**answer

194 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

**2**answers

223 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

**1**answer

95 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

**2**answers

271 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 ...