# Tagged Questions

**0**

votes

**1**answer

63 views

### Lower semicontinuity of a Bochner integral of a convex function

I'm looking for the following result:
Let $\Omega \subset \mathbb{R}^n$ be a bounded domain. The map
$$u \mapsto \int_0^T \int_{\Omega} f(u(t))$$
is lower semicontinuous for $u \in ...

**2**

votes

**0**answers

139 views

### Showing $\langle \frac{\partial b(v)}{\partial t}, v \rangle_{H^{-1}(\Omega), H^1(\Omega)} = \frac{d}{dt}\int_{\Omega}\Psi^*(b(v))$

Let $b$ be continuous and increasing with $b(0) = 0$. Define $\Psi(t) = \int_0^t b(s)\;ds$ and $\Psi^*(s) = \sup_{r \in \mathbb{R}} (sr-\Psi(r))$.
(Note $\Psi^*(b(s)) + \Psi(s) = sb(s)$).
Let $v ...

**1**

vote

**0**answers

72 views

### convergence of supergradient

Let $\{g_n\}$ be a sequence of concave functions defined on $\mathbb{R}$ and set
$$\lambda_n(x)=\lim_{\Delta x\to 0+}\frac{g_n(x+\Delta x)-g_n(x)}{\Delta x}$$
Assume there exists a concave function ...

**0**

votes

**0**answers

36 views

### convergence of concave envelope

Let $\{f_n\}$ be a sequence of uniformly upper bounded functions defined on $\mathbb{R}$ s.t. for every $x\in\mathbb{R}$
$$f_n(x)\to f(x),~ n\to\infty$$
Define $g_n$ and $g$ as the concave envelope ...

**0**

votes

**1**answer

63 views

### Extending affine maps defined on weakly closed sets to the whole topological space

Given $C$ a weakly closed convex subset of a (real) Banach space $B$, with $0\in C$ and $\varphi:C\longrightarrow \mathbb{R}$ weakly continuous, with $\varphi(0)=0$, can we extend $\varphi$ to a ...

**3**

votes

**1**answer

157 views

### Characterization of a set in $\mathbb{R}^d$

Let $X= (X_1,\dots, X_d)$ be a fixed vector of random variables on the space $(\Omega, \mathcal{F}, \mathbb{P})$. Consider the following set.
\begin{equation}\label{main12}
C= \{x\in \mathbb{R}^d ~|~ ...

**-1**

votes

**1**answer

85 views

### 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 ...

**0**

votes

**1**answer

132 views

### Exponential Convexity

$\textbf{Definition:}$ 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 ...

**1**

vote

**1**answer

196 views

### positive semigroups and convex operator

Let $\{Z(t)\}_{t\geq 0}$ be a strongly continuous positive semigroup on a Banach lattice V. (endowed with ordering $\leq$). $\phi:V\rightarrow V$ is a convex operator. I want to prove that ...

**0**

votes

**0**answers

97 views

### Uniform convergence of difference quotients of a convex function

Let $f(\cdot):\mathbb{R}^n\rightarrow \mathbb{R}$ be a convex function. At $x$ denote the subdifferential by $\partial f(x)$ which is compact and closed. Now, define the approximation of $f$ around a ...

**2**

votes

**1**answer

360 views

### existence of a minimum for a convex functional on a non-reflexive space

Let $X$ be a Banach space; $K\subset X$ nonempty, closed and convex; and $f:K\to \mathbb R$ lower semicontinuous, convex functional. Let also $f$ be coercive, i.e., $f(x)\to +\infty$ as $\|x\|\to ...

**3**

votes

**3**answers

323 views

### When is a sequentially closed cone, closed?

The following question I also posed here, but still got no answer. Let $X$ be a locally convex, Hausdorff topological vector space and $C\subseteq X$ a convex cone, which is sequentially closed. What ...

**3**

votes

**0**answers

152 views

### Ways to establish equality of measures on locally compact spaces

Let $M$ be a locally compact space and $\mu$ be some probability measure on $M$. Let $y^\ast \in M$, $f(x,y)$ be a real continuous bounded function $M \times M \to \mathbb{R}$. Consider an equality
$$
...

**2**

votes

**1**answer

616 views

### Projection exists => Uniformly convex?

Hello,
I know that: Let X be a uniformly convex Banach-Space, $a\in X$ and $C\subset X$ closed and convex, then there is a unique $b\in C$ with $\left\Vert a-b\right\Vert=\inf_{x\in C}\left\Vert a-x ...

**0**

votes

**0**answers

123 views

### variational problem under convexity constraints

I wonder if there is any method to compute variational problems subject to certain shape constraints (e.g., convexity, monotonicity, etc.). The literature I found on this topic (which I am no expert ...