Intereting Posts

Evaluation of limit at infinity: $\lim_{x\to\infty} x^2 \sin(\ln(\cos(\frac{\pi}{x})^{1/2}))$
How to prove that if a prime divides a Fermat Number then $p=k\cdot 2^{n+2}+1$?
Finding the n-th lexicographic permutation of a string
Why only two binary operations?
which of the following is NOT a possible value of $(e^{f})''(0)$??
Proof that this is independent
If $0<a<b$, prove that $a<\sqrt{ab}<\frac{a+b}{2}<b$
Finding the Radius of a Circle in 3D Using Stokes Theorem
Besov–Zygmund spaces and the Inverse Function Theorem, is the Inverse Zygmund?
if locally Lipschitz implies Lipschitz on compacts.
Subgroup generated by a set
Characterization of an invertible module
When the $a^2+b+c+d,b^2+a+c+d,c^2+a+b+d,d^2+a+b+c$ are all perfect squares?
A family having 4 children has 3 girl children. What is the probability that their 4th child is a son?
Is $n \sin n$ dense on the real line?

I have been stumped for long by this exercise (3.12(d)) from Stokey and Lucas’s *Recursive Methods in Economic Dynamics*. Would greatly appreciate any hints.

Let $\phi: X \to Y$ and $\psi: X \to Y$ be lower hemicontinuous correspondences (set-valued functions), and suppose that for all $x \in X$

$$\Gamma(x)=\{y \in Y: y \in \phi(x) \cap \psi(x)\}\neq \emptyset$$

- Continuity of a convex function
- Is the composition of $n$ convex functions itself a convex function?
- Dual of a rational convex polyhedral cone
- Projection and Pseudocontraction on Hilbert space
- If $f$ is continuous and $\,f\big(\frac{1}2(x+y)\big) \le \frac{1}{2}\big(\,f(x)+f(y)\big)$, then $f$ is convex
- Dual to the dual norm is the original norm (?)

Show that if $\phi$ and $\psi$ are both convex valued, and if $\mathrm{int} \phi(x) \cap \mathrm{int} \psi(x) \neq \emptyset$, then $\Gamma(x)$ is lower hemicontinuous at $x$.

[A correspondence $\Gamma: X \to Y$ is said to be *lower hemicontinuous* at $x \in X$ if $\Gamma(x)$ is nonempty and if, for every $y \in \Gamma(x)$ and every sequence $x_n \to x$, there exists $N \geq 1$ and a sequence $\{y_n\}_{n=N}^\infty$ such that $y_n \to y$ and $y_n \in \Gamma(x_n)$, all $n \geq N$.

Intuitively this means that the graph of $\Gamma(x)$ cannot suddenly broaden out.]

**EDIT**: We can assume that $X$ and $Y$ are subsets of $\mathbf{R}^n$.

- Legendre transform of a norm
- Operations research book to start with
- Is this set compact?
- DE solution's uniqueness and convexity
- The Proximal Operator of the $ {L}_{\infty} $ (Infinity Norm)
- On the convexity of element-wise norm 1 of the inverse
- Limit of CES function as $p$ goes to $- \infty$
- mid-point convex but not a.e. equal to a convex function
- Prove that convex function on $$ is absolutely continuous
- Every closed convex cone in $ \mathbb{R}^2 $ is polyhedral

Here is a somewhat detailed outline of an argument that I think works. As this is a homework problem, some of the pieces of the argument do need to be filled in.

Assume we’re in $\mathbb{R}^m$. For fixed $y \in \Gamma(x)$, the fact that $\Gamma(x)$ has a nonempty interior means that there are $m$ points $z_1, z_2, \ldots, z_m$ in the interior of $\Gamma(x)$ such that these $m$ points and $y$ together are affinely independent. Thus you can take sufficiently small balls around each of these $m+1$ points such that the balls do not intersect and that any set consisting of one point from each ball is also affinely independent. Let $z_0 = y$. Since $\phi$ is lower hemicontinuous, for each $z_i$ there exists a sequence $z_{i_n} \to z_i$ and an $N_i$ such that $z_{i_n} \in \phi(x_n)$ and $z_{i_n}$ is inside that small ball around $z_i$ for all $n \geq N_i$. For each $n \geq \max \{N_i\}$, construct the convex hull $C_n$ of $\{z_{0_n}, z_{1_n}$, $z_{2_n}, \ldots, z_{i_n}\}$. Since $\phi$ is convex-valued, $C_n$ is a subset of $\phi(x_n)$. Do the same thing for each $n$ for the $\psi$ function to obtain sets $D_n$. Let $S_n = C_n \cap D_n$. The intersection of the convex hulls of two sets of $m+1$ affinely independent points in $\mathbb{R}^m$ that are pairwise close to each other must be nonempty. (Consider the supporting hyperplanes.) Let $y_n$ be the point in $S_n$ closest to $y$. Since the extreme points of $S_n$ converge to the extreme points of the convex hull of $\{z_0, z_1, z_2, \ldots, z_m\}$, $S_n$ converges as a set to the convex hull of $\{z_0, z_1, z_2, \ldots, z_m\}$. Thus the point in $S_n$ closest to $y$ $(= z_0)$ must converge to $y$; i.e., $y_n \to y$.

- Given that $p$ is a prime and $p\mid a^n$, prove that $p^n\mid a^n$.
- Find smallest number which is divisible to $N$ and its digits sums to $N$
- Divergent series and $p$-adics
- Can multiplication be defined in terms of divisibility?
- Expected value of integrals of a gaussian process
- If a Laplacian eigenfunction is zero in an open set, is it identically zero?
- Gathering books on Lorentzian Geometry
- Calculating $\sum_{n=1}^\infty\frac{1}{(n-1)!(n+1)}$
- What does “curly (curved) less than” sign $\succcurlyeq$ mean?
- Is there a difference between $y(x)$ and $f(x)$
- Quaternions vs Axis angle
- Integrable – martingale
- Spectrum of infinite product of rings
- Continuous and bounded variation does not imply absolutely continuous
- Hitting times for Brownian motions