The convexity tag has no wiki summary.

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### Quantitative stability: Hausdorff distance between subdifferentials

Suppose I have two convex functions $f$ and $g$ mapping $\mathbb{R}^{n}\to\mathbb{R}$ (so they are 'more' than proper). Suppose $\|f-g\|_{\infty}<\epsilon$, the sup-norm. In particular, the ...

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54 views

### Regularity of the Minkowski functionnal of a convex

Let $K$ be a convex compact set in $\mathbb{R}^2$ with $0 \in \overset{\circ}{K}$. The Minkowski functional associated to K is:
\begin{align*}
\varphi_K(x):= \inf \left\{t>0 \; : \; tx \in K ...

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98 views

### Can we define log-convex operators?

Let $I\subset\mathbb{R}$. A function $f:I\rightarrow\mathbb{R}$, is said to be log-convex if $\log f$ is convex or equivalently for all $x,y\in I$ and $\alpha\in [0,1]$
$$f(\alpha x+(1-\alpha)y)\leq ...

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56 views

### How is this transformation related to the Legendre transform?

I stumbled over the following transform in a statistical mechanics paper:
Unfortunately, no mathematical details were given there, which is why I wanted to ask here about this transform.
Let $s : ...

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224 views

### Convex Sets and Nearest Neighbors

For a set $S \subseteq \mathbb{R}^n$ and a point $x \in \mathbb{R}^n$, let $c_S(x)$ be the point $s \in S$ that minimizes $\|x-s\|$ if such a point exists and is uniquely determined. It is known that ...

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**1**answer

141 views

### Is there a non-compact Poulsen simplex?

A Choquet simplex is a closed, convex and metrizable subset of a locally convex Hausdorff topological vector space in which every point is a barycenter of a unique probability measure supported on the ...

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56 views

### Does this squared distance functional have a unique critical point on geodesically convex manifolds?

Let $M$ be a Riemannian manifold with distance function $d$, $C \subset M$ a geodesically convex set, $a=(a_i)_{i=1}^n \in C^n$, $W \in \mathbb{R}_{\geq 0}^{n \times n}$ and $J\colon C^n \rightarrow ...

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202 views

### Characterization of closed convex set

Suppose that $C$ is a bounded convex subset of $\mathbb{R}^n$ such that the optimum value, over $C$, of any linear functional is attained at some point of $C$. Does this imply that $C$ is closed? If ...

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93 views

### Zariski closure of the boundary of a closed convex subset of ${\mathbb R}^n$

Let $\eta$ be a closed convex subset of ${\mathbb R}^n$ of convex dimension $n$ (not necessarily compact) with nonempty boundary $\partial \eta$. Then $\eta$ is an $n$-dimensional topological ...

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78 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) ( ...

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104 views

### Distance from constant width bodies

EDIT As @David has observed, my conjecture was clearly wrong for $\ n:=2.\ $ Let me still give it a chance for $\ n\ge 3$.
I'll call a family $\ F\ $ of bound closed convex subsets of $\ \mathbb ...

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**1**answer

57 views

### Concavity of the solution of a parametric implicit function

Suppose $F(x,y;k)=f(x,y)+kg(x,y)=0$ uniquely defines the solution $y(x;k)$ for $x\in \mathbb{D}$, a compact domain, and $0\leq k \leq 1$ is a parameter. We know that for $k=0,1$, $y(x;0)$ and $y(x,1)$ ...

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98 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 ...

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120 views

### When does a cone contain its dual cone?

Let $V$ be a finite-dimensional vector space with an inner-product $(,)$ and let $C\subset V$ be a cone in $V$. Let $C^\vee$ denote the dual of $C$ with respect to $(,)$, i.e., the set of vector $v\in ...

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71 views

### Convex Optimization related problem

Suppose two non-negative convex functions $f$ and $g$ be given.
We want to solve the following optimization
$$\max_{g\leq\epsilon}f.$$
Now suppose that both $f$ and $g$ can be upper-bounded by a ...

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23 views

### Changing a nonlinear equality constraint into some conic inequality plus rank constraint

If we have a constraint optimization problem in which one of our constraint is $\prod\limits_{k = 1}^N {\left( {x - {a_k}} \right) = 0} $ . How could this nonlinear equality condition be changed into ...

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69 views

### Is first term of my cost function convex?

I have an optimization problem in the form of
[\begin{array}{l}
\mathop {{\rm{Minimize}}}\limits_{\bf{X}} \,\,\,2\left| \delta \right|\sqrt {{\rm{Tr}}\left( {{\bf{A}}{{\bf{X}}^2}} \right)} {\rm{ - ...

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337 views

### Are all linear transformations measurable?

Let $V$ and $W$ be topological vector spaces over $\mathbb{F}$ (with $\mathbb{F}=\mathbb{R}$ or $\mathbb{C}$), and let $T:V \to W$ be a linear transformation. It is well-known that $T$ is not ...

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117 views

### Convexity of a certain sublevel set

Consider the polynomial of degree $4$ in the variable $r$ :
$$ r^4 + (x^2 + y^2)\ r^2 - 2 x y\ r + x^2 y^2 $$
The discriminant of this polynomial in $r$ is the following expression
(obtained using ...

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107 views

### Reference request: Continuity of unique maximizer of linear functional on convex set

Does anyone know reference for a theorem of the following sort:
Proposition: Let $K \subset\mathbb {R}^n$ be a compact convex set, and assume that
$$f(w):=\operatorname{argmax}_{x\in K}w(x) $$ is ...

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193 views

### An inequality concerning convexity and expectation

Assume $f$ and $g$ are nonnegative with
$$\int_0^\infty f(x)dx=1=\int_0^\infty g(x)dx
$$
and
$$\int_0^\infty xf(x)dx<\infty > \int_0^\infty xg(x)dx
$$
Is it true for nonnegative numbers $p$, $q$ ...

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113 views

### Is this graph of reciprocal power means always convex?

Let
$$
p = (p_1, \ldots, p_n)
$$
be a finite probability distribution, which for convenience I'll assume to have no zeroes: thus, $p_i > 0$ for all $i$ and $\sum_i p_i = 1$.
Is the function
...

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41 views

### Convergence rate of stochastic gradient decent with projections

Given a strong (not only strict) convex function $f: \mathbb{R}^n\to\mathbb{R}$. On such problems, stochastic gradient decent (SGD) has a convergence rate of $O(1/T)$, where $T$ is the number of ...

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99 views

### Convexity of the Frobenius norm of the product of two matrices

I have the following function for two matrices ${\bf A}$ and ${\bf B}$:
$f({\bf A}, {\bf B}) = \| {\bf Y - XAB} \|_F^2 = trace\{({\bf Y - XAB)}^T({\bf Y - XAB)}\}$
where matrices ${\bf X}_{n \times ...

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83 views

### Lower convex envelope of a function involving entropy

Suppose two discrete random variables $X$ and $Y$ defined on finite sets $\mathcal{X}$ and $\mathcal{Y}$ are given and also suppose the conditional distribution $P_{Y|X}$ (i.e, channel) is fixed. We ...

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### Finite union of closed convex sets is triangulable?

I posted this question on Stackoverflow, but didn't get an answer.
Let $A_1, \ldots, A_k \subseteq \mathbb{R}^n$ be closed convex sets. Is the union $\bigcup_{i=1}^k A_i$ triangulable, that is, ...

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77 views

### proving quasi convexity of multivariable function

Given
an arbitrary $(N \times N)$ square matrix ${\bf X}$
a positive definite $(M\times M)$ matrix ${\bf T}$
a $(Q\times MN), Q< MN$ matrix ${\bf Z}$ consisting of only 1s and 0s where there is
...

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153 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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68 views

### Covering fat objects with fat objects

The family of rectangles has the cover property, i.e.:
For every $R\geq 1$, $k\geq 1$: every rectangle with aspect ratio $kR$ can be exactly covered by $\lceil k\rceil$ (possibly overlapping) ...

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### Strictly Convex Smoothing of a function defined on an affine manifold

A function defined on an interval is strongly convex if it has positive definite second derivative. A function on an affine manifold is strongly convex
if its restriction to each line is. An affine ...

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106 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 ...

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69 views

### Closed approximately convex set is convex

In Convex Analysis and Nonlinear Optimization by Borwein and Lewis there's the following exercise, stating that a closed approximately convex is convex:
...

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85 views

### Is an integral against a probability measure in the convex hull of the range?

This may be really obvious but I don't see it. Let $f:\Omega \to \mathbb R^n$ be integrable with respect to a probability measure $\mu$. Does it follow that $\int_\Omega f \, d\mu$ is in the convex ...

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246 views

### What it is the volume of the unit ball section of the cone of positive definite matrices?

Let $PD_{n}$ be the cone of positive definite $n \times n$ real matrices and let $B$ be the unit sphere in $n \times n$ dimensions. What is the volume of $PD_{n} \cap B$?
EDIT: Let's assume that $B$ ...

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81 views

### additive support functions of a convex set

Let $K \subset \mathbb{R}^d$ be a compact, convex set. It could be uniquely determined by its support function (for $u$ on the unit-sphere $S^{d-1}$), given by
$$h_K(u) = \sup \{ \sum_{i=1}^d x_i ...

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76 views

### Is there a name for functions f(x,y) that are only convex in x (and continuous in y). [closed]

This is just a notation question.
If I cannot find a preexisting name, I would try one-sided convex, or something of the sort.

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145 views

### Question about the log-det function

Suppose I have a diagonal $n \times n$ matrix $\Gamma$ with positive entries, and a fixed $n \times k$ matrix $P$ with $P^\intercal P = I$ (here, $k \leq n$). I'm interested in knowing whether the ...

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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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134 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 ...

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415 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 ...

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### 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 ...

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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 ...

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367 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 ...

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208 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) + ...

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### 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 ...

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119 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 ...

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47 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 ...

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269 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 $g$, ...

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### 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 ...

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195 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 ...