# Questions tagged [convexity]

For questions involving the concept of convexity

600
questions

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### Show this curve is convex

Consider the curve given by $x^4-x^3y+6x^3+x^2y^2 -2x^2y +11x^2-xy^3 -2xy^2 -3xy +6x +y^4+6y^3 +11y^2 +6y = 120A$ where $A$ is a positive integer. This curve is convex, and I know that to show that ...

10
votes

1
answer

312
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### Convex functions in convex sets

Suppose $\Omega \subset \mathbb{R}^n$ is some bounded, convex set. For which domains $\Omega$ is it true that for every convex function $f:\Omega \rightarrow \mathbb{R}$ the average of the function in ...

3
votes

2
answers

82
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### Subdifferential of a convex function admits a continuous selection

Let $F$ be a continuous convex function on $\mathbb{R}^n$.
If the subdifferential $\partial F(x)$ of $F(x)$ admits a continuous selection, for every $x \in \mathbb{R}^n$, does it mean that $F$ is ...

2
votes

1
answer

79
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### Submodularity of a particular function derived from a convex function?

Consider a convex function $f : \mathbb{R}^d \to \mathbb{R}$. Define now the set-input function $g_f : 2^{[d]} \to \mathbb{R}$ as follows,
\begin{align}
g_f(S) = \min \left\{ f(x) : x \in \mathbb{R}^d ...

3
votes

1
answer

358
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### An exercise on log-concave random variable on the real line

Let $X$ be a real random variable with log-concave density $f$. Assume that $E(X) =0$ and $E(X^2)=1$.
Show that there is a universal (independent of $X$) constant $c>0$ such that:
$$P(X\in[-1/2;0])\...

0
votes

0
answers

45
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### Representation of concave point-to-set maps

Given a point-to-set map $C: X \rightrightarrows Y$ defined by some vector valued-function $\mathbf{g}: X \times Y \to \mathbb{R}^n$ such that $C(x) \doteq \{y \in Y | g_1(x,y), …, g_n(x,y) \geq 0 \}$,...

2
votes

1
answer

98
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### Tangent cone of a closed convex cone

Let $K \subset \mathbb{R}^n$ be a closed convex set. Given a point $u \in K$, the tangent cone of $K$ at $u$ is defined as (or characterized by)
$$
T_K(u) := \mathrm{cl}(\left\{ t (v - u) \mid v \in K,...

0
votes

0
answers

84
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### Does there exists a smooth reflexive infinite dimensional Banach space that is not strictly convex

It is known that in a reflexive Banach space, if the norm is strictly convex, then its dual will be smooth Banach space, and if the norm is smooth, then the dual norm is strictly convex.
We can find ...

0
votes

1
answer

99
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### Smoothness of a Hilbert space under an equivalent norm

Let us take the Hilbert space $l_2$ with an equivalent norm
$\Vert x \Vert = \max \{2 \Vert x \Vert_1, \Vert x \Vert_2 \}$, where $\Vert x \Vert_1 =( \sum_{n=2}^\infty x_n^2 )^{\frac{1}{2}}$ and $\...

4
votes

2
answers

112
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### 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 ...

0
votes

0
answers

47
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### Is the absolute value of a complex quadratic form a convex real function?

Consider a complex vector space $V = \mathbb{C}^n$ and a quadratic form $Q(x) = x^TAx$ on $V$ where $A$ is a symmetric matrix i.e., $A^T = A$.
Is is true that the absolute value $|Q(x)|$, seen as a ...

1
vote

2
answers

80
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### Establishing quasiconcavity

Let $f:\mathbb{R}_+\rightarrow\mathbb{R}_+$ be twice differentiable quasi-concave function satisfying $f(x)>0,\forall x \in \mathbb{R}_+$. Let $g:\mathbb{R}_+\rightarrow\mathbb{R}_+$ be a positive, ...

4
votes

1
answer

104
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### Approximation of convex bodies by polytopes corresponding to smooth toric varieties

Let $P\subset \mathbb{R}^n$ be an $n$-dimensional polytope with rational vertices. There is a well known construction which produces an $n$-dimensional algebraic variety $X_P$ called toric variety. In ...

2
votes

1
answer

137
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### Higher-order convexity

Let $f \in C^\infty(\mathbb R)$.
$f^{(1)}=f'\geq 0$ iff $\forall (a,b) \in\mathbb R^2,a\leq b$ then $f(a) \leq f(b)$
$f^{(2)}=f''\geq 0$ iff $\forall (a,b)\in\mathbb R^2,\forall t\in [0,1], f(ta+(1-t)...

1
vote

1
answer

54
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### Any example of a multi-valued monotone maximal operator without subdifferential?

Is there an example of a multivalued maximal monotone operator that is not the convex subdifferential of a proper convex lower semicontinuous? Besides, among these type of operators, are there any ...

11
votes

2
answers

517
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### Existence of an open convex set

Let $T$ be a normed vector space, $K\subseteq T$ compact and convex and $O\subseteq K$ convex and open in $K$ (i.e. open w.r.t. the subspace topology of $K$ inherited by $T$).
Can we find an open set $...

1
vote

1
answer

128
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### Is a Lipschitz continuous gradient equivalent to this condition?

I know if a function $f: \mathbb{R}^n \to \mathbb{R}$ is $L$-smooth, i.e. its gradient $\nabla f$ is $L$-Lipschitz continuous, then it satisfies the following inequality for any $x, x_0 \in \mathbb{R}^...

0
votes

0
answers

42
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### On the convex inequality $x^2\leq u$

Suppose $x$ is in $[0,M]$ where $M^2$ is known and treated as constant in a convex program.
Consider the convex inequality $x^2\leq u$. The reverse inequality $u\leq x^2$ is non-convex. On the other ...

4
votes

0
answers

125
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### 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{...

5
votes

1
answer

135
views

### On existence of a concave function

Let $a$ be a strictly positive $C^\infty$ smooth function on the unit interval. Does there exist a strictly positive $C^\infty$ smooth function $f$ on $I$ such that
$$ f’’(x) \leq 0\quad \text{and} \...

1
vote

0
answers

47
views

### Comparison of solutions of Hamilton-Jacobi equations with different initial conditions

Consider a Hamilton-Jacobi equation:
$$u_{t} + f(u_{x}) = 0 \quad (x,t) \in \mathbb{R}\times [0,+\infty)$$
with two possible initial conditions $u(x,0) = g_{i}(x)$ for $x \in \mathbb{R}$ and $i=1,2$. ...

0
votes

1
answer

58
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### Do subgradient inequalities hold for matrix convex functions?

Suppose $f$ is a matrix convex function over symmetric, positive semidefinite matrices with spectra in some interval $I$ [1]. That is, for $A,B\succeq 0$ with spectra in $I$, and any $\theta\in[0,1]$,
...

0
votes

1
answer

71
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### On the measure of nonconvexity (MNC)

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\operatorname{conv}(A)} \...

7
votes

0
answers

323
views

### 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:...

1
vote

1
answer

110
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### Are convex functions on manifolds the same as $c$-convex functions, where $c(x,y)=d(x,y)^2/2$?

I am reading the following book on optimal transport. While reading I came across the following definition of $c-$convexity.
Given $X$ and $Y$ metric spaces, $c: X \times Y \rightarrow \mathbb{R}$, ...

0
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0
answers

42
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### "More" cyclical monotonicity

Let $X$ and $Y$ be some finite sets. For a given function $f:X\times Y\rightarrow \mathbb{R}$, we say a set $S\subset X\times Y$ is $f$-cyclically monotone if for any sequence $(x_1,y_1),...,(x_n,y_n)\...

2
votes

2
answers

166
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### Elementary convexity example

I'm trying to check that certain examples of Young functions in the harmonic analysis literature are actually Young functions, and in doing so need to prove the following convexity-like inequality for ...

0
votes

1
answer

57
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### Strict inclusion for recession cone of closure of a convex set

Let $C$ be a nonempty closed convex subset of $\mathbb{R}^n$. The recession cone of $C$ is given by
$$R_C=\left\lbrace d\in\mathbb{R}^n:x+td\in C, \forall t>0, \forall x\in C\right\rbrace.$$
It is ...

1
vote

0
answers

127
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### The image of zero-measure set under normal mapping is Lebesgue measurable

Let $u$ be a convex function defined on a bounded open set $\Omega$ in $\mathbb{R}^n$. Then $u$ is twice differentiable a.e. Let $E_u$ be the set on which $u$ is not twice differentiable. Then $E_u$ ...

6
votes

2
answers

201
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### For most directions does the supporting hyperplane meeting a bounded convex set meet it in one point?

Let $C\subseteq \mathbb R^n$ be non-empty, convex and compact. For $v\in S^{n-1}$, let $H_v$ be the supporting hyperplane in the direction of $v$ (i.e., $H_v$ is the boundary of the smallest closed ...

18
votes

3
answers

652
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### Convergence of convex functions

I can prove the following result.
Theorem 1. Let $f_n:\mathbb{R}^n\to \mathbb{R}$ be a sequence of convex functions
that converges almost everywhere to a function $f:\mathbb{R}^n\to\mathbb{R}$.
Then ...

7
votes

0
answers

103
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### 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\...

0
votes

0
answers

123
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### Lipschitz map on positive definite cone of $n$-by-$n$ matrices

A function matrix $f : X \to \mathbb R$ is a convex Lipschitz continuous matrix function with Lipschitz constant $\mathrm L$ with respect to a fixed given norm $\|\cdot\|$, i.e., $|f(A)-f(B)| \leq \...

2
votes

1
answer

249
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### Intersection of the simplex with a linear subspace of codimension $2$

The sets are defined in $\mathbb{R}_+^n$ $(n\geq 1)$. The relative interior of a convex set $C$ is denoted $\mathring C$.
Let $S$ be the $n$-simplex:
$$S=\left\{x\in\mathbb{R}_+^n,\,\sum_{i=1}^n x_i=1\...

2
votes

1
answer

99
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### Can we use the solution to two optimisation problems to solve a third, bigger, one?

Background
Say we have an optimization problem $$\min_x f(x) = g(x) + h(x)$$
where $g$ is differentiable and convex, and $h$ are convex but not necessarily differentiable. If $g$ is the mean squared ...

0
votes

0
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45
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### Intersection of the simplex with a space vector

The sets are defined in $\mathbb{R}_+^n$ $(n\geq 1)$. The relative interior of a convex set $C$ will be denoted $\mathring C$. The codimension of a submanifold $M$ of a manifold $N$ will be denoted ...

1
vote

1
answer

123
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### A question on convexity and conjugate points

Let $(M,g)$ be a compact smooth simply connected Riemannian manifold with a smooth boundary. Assume also that $(M,g)$ does not have any conjugate pairs of points. Let $\Gamma \subset \partial M$ be a ...

2
votes

1
answer

174
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### Concavity/convexity of distance-to-boundary function

For $\Omega$ a bounded open set of $\mathbf{R}^d$, denote $\mathrm{d}_\Omega:x\mapsto \mathrm{d}(x,\partial\Omega)$ the distance-to-boundary function.
If $\Omega$ is convex, a short argument recalled ...

0
votes

2
answers

191
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### Decreasing magnitude of spherical centroid

Let $\sigma$ be the uniform measure on $\mathbb{S}^{d-1}\subset \mathbb{R}^d$. For any region $R\subset \mathbb{S}^{d-1}$, let $X_R$ be a random variable which is uniformly distributed across $R$. We ...

0
votes

0
answers

68
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### Literature request: proving or disproving convexity of the optimal value function of semidefinite program (SDP) or convex optimization in general

Suppose I have a function $f:\mathbb{R}\rightarrow \mathbb{R}$ defined as the following parametric optimization problem:
$$f(p) = \inf_xf_0(x) \quad \text{subject to } \quad G(x,p)\leq 0,$$
where ...

5
votes

1
answer

210
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### Proving the set $\lbrace \frac{(x + y)^2}{\sqrt{y}} \leq x - y + 5, y > 0 \rbrace$ is convex

I have recently picked up a course on Convex Analysis in my spare time, but feel I'm not quite up to speed with the 'tricks' for proving a set is convex.
I have managed to prove this by moving all ...

2
votes

1
answer

318
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### Are there "pathological convex sets" over ultravalued fields of char 2?

In their book Topological Vector Spaces (2nd ed.) Lawrence Narici and Edward Beckenstein generalise convex sets for TVS over ultravalued field $K$ as $K$-convex sets. The definition goes as following:...

1
vote

1
answer

135
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### Lipschitz aspect of a projection on the boundary of a convex

Let $C$ be a closed convex set of $\mathbb{R}^n$ $(n\geq 1)$, with asymptotic cone $C^{as}$ having for interior $\text{Int}\big(C^{as}\big)$. Let $u\in\mathbb{R}^n\setminus\{0\}$ such that
\begin{...

4
votes

2
answers

322
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### Is this projection on the boundary of a convex Lipschitz?

Let $C$ be a closed convex set of $\mathbb{R}^n$ $(n\geq 1)$, and $u\in\mathbb{R}^n\setminus\{0\}$ such that $u$ does not belong to the asymptotic cone of $C$ and is nowhere tangent to $\partial C$. ...

5
votes

0
answers

95
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### 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 ...

1
vote

1
answer

96
views

### Intersecting points of increasing convex functions

Can two increasing and differentiable convex functions agree exactly on a countable set of cardinality greater than two?

3
votes

2
answers

194
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### Hausdorff dimension of the non-differentiability set a convex function

Let $X \subset \mathbb R^d$ be open, $f : X \to \mathbb R$ and
$$
E := \{x \in X : f \text{ is not Fréchet differentiable at }x\}.
$$
Then we have the following result which is
Theorem: If $X= \...

1
vote

1
answer

60
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### Volume ratio of polytopes with few vertices

The volume ratio of a convex body $K\subset \mathbb{R}^{n}$ is $v_r(K) = \inf_{\mathcal{E}\subset K} \left(\frac{Vol(K)}{Vol(\mathcal{E})}\right)^{1/n}$ where the infimum run over ellipsoids included ...

2
votes

1
answer

139
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### Gradient of a convex function on $\mathbb{R}^d$, maximum on hypercubes bounded by values in corners?

Let $f : \mathbb{R}^d \rightarrow \mathbb{R}$ be infinitely often continuously differentiable and convex.
For $d = 1$, we know that for any interval $[a, b]$, it holds for $x, y \in [a, b]$ that
$$
(f'...

6
votes

1
answer

272
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### Convex solutions of the Poisson equation

Let $D$ be a planar, bounded, convex open domain. Given a positive function $f:D\to(0,+\infty)$, let us consider the Poisson equation
$$\Delta u=f\quad\hbox{in }D.$$
Not specifying any boundary ...