Questions tagged [convexity]

For questions involving the concept of convexity

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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 ...
Benjamin Warren's user avatar
10 votes
1 answer
312 views

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 ...
Stefan Steinerberger's user avatar
3 votes
2 answers
82 views

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 ...
Aimar's user avatar
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2 votes
1 answer
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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 ...
Television's user avatar
3 votes
1 answer
358 views

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])\...
Gericault's user avatar
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0 answers
45 views

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 $C(x) \doteq \{y \in Y | g_1(x,y), …, g_n(x,y) \geq 0 \}$,...
Ded's user avatar
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2 votes
1 answer
98 views

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,...
aest's user avatar
  • 133
0 votes
0 answers
84 views

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 ...
Priyanka Priyadarshini Behera's user avatar
0 votes
1 answer
99 views

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 $\...
Priyanka Priyadarshini Behera's user avatar
4 votes
2 answers
112 views

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 ...
Mikhail Tikhomirov's user avatar
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0 answers
47 views

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 ...
user493645's user avatar
1 vote
2 answers
80 views

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, ...
oyy's user avatar
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4 votes
1 answer
104 views

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 ...
asv's user avatar
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2 votes
1 answer
137 views

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)...
Dattier's user avatar
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1 vote
1 answer
54 views

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 ...
megaproba's user avatar
  • 351
11 votes
2 answers
517 views

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 $...
Julian's user avatar
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1 vote
1 answer
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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}^...
aest's user avatar
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0 answers
42 views

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 ...
Turbo's user avatar
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4 votes
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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{...
Motaka's user avatar
  • 291
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} \...
Ali's user avatar
  • 3,863
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 $i=1,2$. ...
IamWill's user avatar
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0 votes
1 answer
58 views

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 $\theta\in[0,1]$, ...
VF1's user avatar
  • 253
0 votes
1 answer
71 views

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)} \...
Motaka's user avatar
  • 291
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:...
Piotr Hajlasz's user avatar
1 vote
1 answer
110 views

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, $c: X \times Y \rightarrow \mathbb{R}$, ...
Student's user avatar
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0 answers
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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)\...
user_XL's user avatar
2 votes
2 answers
166 views

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 ...
Joshua Isralowitz's user avatar
0 votes
1 answer
57 views

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 ...
Chivul's user avatar
  • 29
1 vote
0 answers
127 views

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 $E_u$ ...
User1999's user avatar
6 votes
2 answers
201 views

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 ...
Alexander Pruss's user avatar
18 votes
3 answers
652 views

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 ...
Piotr Hajlasz's user avatar
7 votes
0 answers
103 views

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\...
alesia's user avatar
  • 2,524
0 votes
0 answers
123 views

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 \...
Reza's user avatar
  • 1
2 votes
1 answer
249 views

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\...
G. Panel's user avatar
  • 513
2 votes
1 answer
99 views

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 ...
user19904's user avatar
0 votes
0 answers
45 views

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 ...
G. Panel's user avatar
  • 513
1 vote
1 answer
123 views

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 ...
Ali's user avatar
  • 3,863
2 votes
1 answer
174 views

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 ...
Ayman Moussa's user avatar
  • 2,410
0 votes
2 answers
191 views

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 ...
Aaron Goldsmith's user avatar
0 votes
0 answers
68 views

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 ...
Shih-Chi Liao's user avatar
5 votes
1 answer
210 views

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 ...
AlwaysLost123's user avatar
2 votes
1 answer
318 views

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:...
Nik Pronko's user avatar
1 vote
1 answer
135 views

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{...
G. Panel's user avatar
  • 513
4 votes
2 answers
322 views

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 $\partial C$. ...
G. Panel's user avatar
  • 513
5 votes
0 answers
95 views

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 ...
guest61's user avatar
  • 319
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?
Sounak's user avatar
  • 15
3 votes
2 answers
194 views

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= \...
Akira's user avatar
  • 713
1 vote
1 answer
60 views

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 ...
Gericault's user avatar
  • 235
2 votes
1 answer
139 views

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'...
Steve's user avatar
  • 1,055
6 votes
1 answer
272 views

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 ...
Denis Serre's user avatar
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