The convexity tag has no usage guidance.

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### (probably simple) optimization question

Suppose you have a concave function defined over a non-polyhedral convex cone and you are interested in the infimum. What would be standard approaches to tackle the question? (The cone is actually PSD ...

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**1**answer

729 views

### Proving that a specific function is quasiconvex

Hello all,
Assume we have a sequence of quasiconcave functions (in $X$) denoted by $f_{i,j}(X)$ for $i,j = 1,\ldots,n$. Denote by $F(X)$ the $n\times n$ matrix whose $(i,j)$ entry is the function $f_{...

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**1**answer

261 views

### Conjecture that two nested convex curves have a point with the same slope

I'm trying to prove a conjecture and need some help.
Consider a continuous, twice differentiable function $p(a)$ such that $p(0) = 0$ and $\forall a$, $p'(a) > 0$ and $p''(a) < 0$ and $p$ is ...

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**1**answer

171 views

### Are faces of a compact, convex body “opposed” iff their extreme points are pairwise “opposed”?

Let $P$ be a compact, convex subset of $\mathbb{R}^n$ (infinite-dimensional generalisations welcome, but not necessary). Let's say that disjoint subsets $W_1$, $W_2$ $\subset P$ are opposed if there ...

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292 views

### Characterising semi-definite positiveness on vectors with non-negative entries

My problem is to characterise (or find useful information on) the cone $C$ of $N\times N$ matrices $M$ ($N\geq 1$) such that $$V^t M V\geq 0$$ for every vector $V $ with non-negative entries. Is this ...

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**1**answer

1k views

### A Weak Form of Borsuk's Conjecture

Problem: Let P be a d-dimensional polytope with n facets. Is it always true that P can be covered by n sets of smaller diameter?
Background and motivation
The Borsuk conjecture (disproved in 1993) ...

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**1**answer

289 views

### Going in the direction of the gradient

First, a motivating example. Suppose $f(x)$ is convex, differentiable, with a single minimum $x^*$.
Then the differential equation $$\dot{x}(t) = -\nabla f(x(t))$$ drives $x(t)$ to $x^*$.
Now my ...

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**1**answer

1k views

### proving the interior of a dual cone is the set of vectors whose inner product is strictly positive on the cone

Apologies for posting such a simple question to mathoverflow. I've have been stuck trying to solve this problem for some time and have posted this same query to math.stackexchange (but have received ...

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**4**answers

522 views

### Packing and isoperimetrics

Suppose a manufacturer bottles small units of liquid and ships them via very large trucks.
If the transportation cost nothing, spherical bottles would minimize the packaging cost (isoperimetric ...

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**1**answer

231 views

### Convexity of a specific semialgebraic set

I have an engineering problem which maybe resolved with semi-definite programming optimization.
I have a set which I would like to know if is convex:
Being $m \in \mathbb{R}^+$ a positive real scalar,...

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302 views

### Deciding the convexity of semialgebraic sets

Given a basic closed semialgebraic set, $S \subset \mathbb{R}^n$, defined by
$S = \{ x \in \mathbb{R}^n \mid g_1 (x) \geq 0 \land \dots \land g_m (x) \geq 0\}$
where $m \in \mathbb{N}$ and $g_1, \...

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832 views

### Is the intersection of boundaries of convex bodies a topological sphere?

Let $K_1, K_2, \ldots K_n$ be convex bodies in $R^d$. Assume that for any index set $I$, $\cap_{i \in I} K_i$ is not empty and is not properly contained in any body $K_i$ for $i\in I$.
Is it true ...

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717 views

### A natural center of a convex weakly compact set in Banach space

Question: Let $S$ be a convex weakly compact set in Banach space $H$. Propose a natural way to define the unique center $O \in S$.
Motivation: A lot! For example, in game theory $S$ can be a set of ...

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**1**answer

361 views

### Quermassintegrals as mean curvature integrals

It is well-known that the quermassintegrals $W_{i}$ of a convex body $K\subset \mathbb{R}^{n}$ can be written as a mean curvature integral
$$ W_{i}(K)=n^{-1}\int_{\partial K}H_{i-1} d\mathcal{H}^{n-1},...

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729 views

### Does the minima of a sequence of convex convergent functions converge?

Suppose $f_1,f_2,\ldots $ is a sequence of convex functions that converges to a continuous convex $f$. Let $x_1^*,x_2^*$ be their respective (not necessarily unique) minima, and let y be a minima of $...

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175 views

### Lipshitz Constant of the convex extension of a submodular function

The title says it all :)
Given a submodular function (take the rank function of a matroid, for a concrete example) $f:\{0,1\}^n\rightarrow \mathbb{R}$, we can extend it to a convex function $g:[0,1]^...

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377 views

### A question regarding Danskin's theorem

Suppose $\phi({\bf x},{\bf z})$ is a continuous function of two arguments,
$\phi: {\mathbb R}^N \times Z \rightarrow {\mathbb R}$,
where $Z \subset {\mathbb R}^m$ is a (non-empty) compact convex ...

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**1**answer

431 views

### Extreme points of a compact convex set are a $G_\delta$?

Dear All,
I'm reading a paper (Residuality of Dynamical Morphisms by Burton, Keane and Serafin) that makes a claim that I've been unable to verify or find a reference for. The claim is made that the ...

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**1**answer

120 views

### Mapping a subset of semi-definite matrices through arcsinus

Hi
I am meeting a problem concerning semi-definite positive matrices, and I have no clue concerning them, the classical approaches I know have not given any result, maybe people used to manipulating ...

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**1**answer

439 views

### Is it possible to extend this inequality about Euclidean distance &Frobenius norm to more general Bregman divergence such as relative entropy & von Neumann divergence?

Motivation- A Special Case
Supposing $A,B\in\mathbb{S}^{m\times m}$ are symmetric positive semi-definite (SPD) matrices and $\mathbf{x}\in\mathbb{R}^m$ is a unit vector where $\|\mathbf{x}\|=1$, we ...

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2k views

### Area Enclosed by the Convex Hull of a Set of Random Points

Consider $n$ points generated randomly and uniformly on a unit square. What is the expected value of the area (as a function of $n$) enclosed by the convex hull of the set of points?

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481 views

### conjugate function for matrix mixed norm

I am familiar with the conjugate function of the vector norm, which uses the concept of dual norm and is defined as follows:
$\|\mathbf{y}\|_p^*=\max_{\mathbf{x}}\left(\mathbf{x}^T\mathbf{y}-\|\...

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227 views

### Is there relationship between $f\left({\sum_i(\mathbf{v}_i^\top\mathbf{x})^2\lambda_i},\sum_j{(\mathbf{u}_j^\top\mathbf{x})^2\theta_j}\right)$ and $\sum_i\sum_j{(\mathbf{v}_i^\top\mathbf{u}_j)^2f(\lambda_i,\theta_j)}$ if $f$ is jointly convex?

Hello, everyone!
As we know that by Jensen's inequality, for jointly convex function $f$ and $\sum_ix_i^2=1$, we have
$$f(\sum_i{x_i^2\lambda_i},\sum_i{x_i^2\theta_i)}\leq\sum_i{x_i^2f(\lambda_i,\...

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268 views

### Generalized unique nearest point problem for a compact, convex set in a strictly convex Banach space.

If $X$ is a Real Banach space with strictly convex norm, it is known that for any non-empty compact, convex set $K$ and point $x_0\notin K$, there exists a unique point $z_0\in K$ minimizing the ...

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### Given a jointly convex function $f$, what is the bound of $f\left(\sum_ip_i^2x_i,\sum_jq_j^2y_j\right)$if $\mathbf{p},\mathbf{q}$ are constrained in a manifold?

Suppose there is a jointly convex function $f:\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}$, $\mathbf{x},\mathbf{y}\in\mathbb{R}^m$ and $\mathbf{p}=[p_1\;\ldots\;p_m]^\top,\mathbf{q}=[q_1\;\ldots\;...

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### Sufficient conditions for gradient descent convergence

I have an unconstrained optimisation problem with convex objective function $f(x)$. Suppose I have access only to some function of the gradient $\hat{\nabla}= g(\nabla f)$, and I take gradient steps ...

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### Computational complexity of unconstrained convex optimisation

What is known about the relationship between unconstrained convex optimisation and computational complexity? For example, for which optimisation problems and which gradient descent algorithms is one ...

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543 views

### Extreme points of unit ball in tensor product of spaces

Let $B_1, B_2$ be unit balls in finite-dimensional normed spaces $X_1, X_2$ respectively.
Let $e(B_1), e(B_2)$ be corresponding extreme points sets.
Consider the unit ball $B$ in tensor product $...

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**1**answer

714 views

### Local strong convexity of a strictly convex function

Is there a formal way to characterise strong convexity about the optimum value of a strictly convex function? I have an objective that looks something like this: $J(p,q) = \sum_{i=1}^{n}d(\pi_{i},q_{i}...

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217 views

### lipschitz property of the derivative of a convex function

Let $f\in C^1(\mathbb R^n\to \mathbb R)$ be a convex function. Suppose the equation
$$f(x+\Delta x)-f(x)-(f'(x),\Delta x) \leq A|\Delta x|^2$$
holds for some constant $A>0$, any $x\in \mathbb R^...

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528 views

### Inequalities for uniformly convex normed spaces

When reading "Chebyshev centers and uniform convexity" by Dan Amir I encountered the following result which is apparently "known and easy to prove". I'm sure it is, but I can't find a proof and am ...

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330 views

### What is the “positive part” of the unit ball in $M_n(R)$ ?

In ${\bf M}_n(\mathbb R)$, let us consider the usual operator norm
$$\|A\|=\sup\frac{\|Ax\|}{\|x\|},$$
where $\|x\|$ is the Euclidian norm.
The closed unit ball $B$ is the set of contractions (in the ...

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505 views

### Tverberg partitions with less than (r-1)(d+1)+1 points

The Tverberg Theorem states the following: Let $x_1,x_2,\dots, x_m$ be points in $R^d$ with $m \ge (r-1)(d+1)+1$. Then there is a partition $S_1,S_2,\dots, S_r$ of $\{1,2,\dots,m\}$ such that $\cap _{...

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284 views

### Erdős-Szekeres empty pseudoconvex $k$-gons

I am wondering if the
Erdős-Szekeres
empty convex $k$-gon question has a different answer if
convexity is replaced by a pseudoline-version of convexity.
The empty convex $k$-gon question
is a variant ...

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**0**answers

107 views

### Mappings preserving convex compactness

Let $H$ be a Hilbert space.
How can one describe continuous mappings $F:H \to H$
that satisfy the following condition:
There exist two elements $c$, $F(c) \neq c$
and a convex compact $M$ containing ...

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### A criterion for the sum of two closed sets to be closed ?

Let $V$ and $I$ be two closed subsets of a Banach space $A$.
The set $V$ is a convex cone, and $I$ is a linear subspace of $A$. I also know that $V\cap I=\{0\}$.
I would like to know whether $I+V$ ...

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**1**answer

502 views

### The maximal “nearly convex” function

The following problem is only tangently related to my present work, and I do
not have any applications. However, I am curious to know the solution -- or
even to see a lack thereof, indicating that the ...

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**1**answer

382 views

### Generalized widths and reverse Urysohn inequalities

This question is inspired by the discussion in MO questions "Local minimum from directional derivatives in the space of convex bodies" and "Bodies of constant width?" about generalized notions of ...

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291 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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### Characterizing curves that bound strictly convex regions

Consider a closed curve on the plane so that if we perform any translation and dilation on it, the resulting curve intersects the original curve at most twice. Does this property characterize the ...

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331 views

### Volume inequality between projections of a convex symmetric set in $\mathbb R^3$

Let $K$ be a centrally symmetric convex body in $\mathbb R^3$ with volume ${\rm vol}(K)=1$. For any subset $F \subset \lbrace1,2,3\rbrace$, let $K_F$ be the projection of $K$ in $\mathbb R^F$.
...

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### Extreme points of a set of probability measures

Consider the set of Borel-measurable probability measures over the interval $[0,1]$ with a given mean, say 1/2. To be precise, I'm talking about the following set $$M=\left(\mu\in \Delta([0,1]):\int x\...

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643 views

### Shift invariant measures that are(n't) convex combinations of ergodic measures

Let $X=2^\omega = \lbrace 0,1 \rbrace^{\mathbb{N}}$ and $T\colon X \to X$ the (left) shift map. The space $\mathcal{M}$ of $T$-invariant Borel probability measures is convex with $\mathcal{M}^e$, the ...

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491 views

### minimizing functions over simple matrix inequalities

I'm wondering if anything is known about minimizing convex, not necessarily linear functions subject to "simple" matrix equalities. To be precise, consider the following example:
$min \Sigma x_i ln ...

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221 views

### Is it possible to solve the argument maximization problem $\arg\max_x \langle x,l \rangle −f_1(x)−f_2(x)$ via convex duality?

I am attempting to solve the argument maximization problem
$$\arg\sup_x \{ \langle x,l \rangle − f_1(x)−f_2(x) \} \ \ \ \ \ \ \ \ \ \ (1)$$
where the functions $f_1$ and $f_2$ are concave but ...

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146 views

### Spline fit with bounded derivations

How can I do a Spline Fit with bounds on some derivations?
Problem
Given:
Set of data points $t_k, x_k$
Set of nodes $n_i$
order $D$ of the spline (in my case $D=5$)
lower and upper bounds $m_d$,$...

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### convexity of images of space-filling curves

Suppose $f:[0,1]\to[0,1]^2$ is continuous and for each $t\in[0,1]$, the area of $\lbrace f(s) : 0\le s\le t \rbrace$ is $t$. For what sets of values of $t\in[0,1]$ can $\lbrace f(s) : 0\le s\le t \...

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1k views

### Does a notion of convex graph make sense?

Let $X=(V,E)$ be a finite connected graph. I would be interested in some notion of convexity.
General question: Is there a notion of convexity for finite connected graphs? How does it look like?
...

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633 views

### Java library for SDP [closed]

People who frequently code semi definite programs, is there any java library for solving sdps? I have tried my luck but all I can find is C/C++ libraries or matlab toolboxes. I can write wrappers to ...

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663 views

### gradient of convex functions

Hello. Can somebody help me with the following question that I have thought over for quite some time, to no avail?
Let $f$ be a smooth function (class $\mathrm{C}^{\infty}$), $f:\mathbb{R}^n \...