# 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 ...
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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 ...
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 ...
79 views

### 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 ...
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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 ...
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1 vote
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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 ...
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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 ...
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1 vote
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### 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 $i=1,2$. ...
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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 $I$ . That is, for $A,B\succeq 0$ with spectra in $I$, and any $\theta\in[0,1]$, ...
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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 ...
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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
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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 ...
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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 ...
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 ...
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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 ...
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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 ...
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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
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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{...
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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 $\partial C$. ...
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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