Let $\mathcal{C}\subseteq\mathbb{R}^d$ be nonempty, closed and convex. Fix some $\mathbf{z}\in\mathbb{R}^d$ and consider: $$ \min_{\mathbf{x}\in\mathcal{C}}\Vert\mathbf{z}-\mathbf{x}\Vert $$ (i) Show that the problem has a solution (ii) Show that the solution is the unique global minimizer   For (i), I have been able to show that $f(\mathbf{x}):=\Vert\mathbf{z}-\mathbf{x}\Vert$ is a convex function. Let $\mathbf{p},\mathbf{q}\in\mathcal{C}$ and $\alpha\in[0,1]$: $$ \begin{align*} f(\alpha\mathbf{p}+(1-\alpha)\mathbf{q}) &amp;= \Vert \mathbf{z} - (\alpha\mathbf{p}+(1-\alpha)\mathbf{q})\Vert \\ &amp;= \Vert \mathbf{z} - \alpha\mathbf{p}-(1-\alpha)\mathbf{q} \Vert \\ &amp;= \Vert \alpha\mathbf{z}+(1-\alpha)\mathbf{z}-\alpha\mathbf{p}-(1-\alpha)\mathbf{q}\Vert \\ &amp;= \Vert \alpha(\mathbf{z} - \mathbf{p})+(1-\alpha)(\mathbf{z} - \mathbf{q}) \Vert \\ &amp;\leq \Vert \alpha(\mathbf{z} - \mathbf{p}) \Vert\ + \ \Vert (1-\alpha)(\mathbf{z} - \mathbf{q}) \Vert \\ &amp;= \alpha\Vert\mathbf{z} - \mathbf{p}\Vert\ +\ (1-\alpha)\Vert\mathbf{z} - \mathbf{q}\Vert \\ &amp;= \alpha f(\mathbf{p}) + (1-\alpha)f(\mathbf{q}) \end{align*} $$ Since $f$ is convex and $\mathcal{C}$ is convex, I have the following necessary and sufficient conditions for optimality (from my lecture notes):  Necessary condition: any local optimal point is stationary. Sufficient condition: if $f$ is convex, any stationary point is global optimal.  So I think that it suffices to show the existence of a stationary point of $f$. Since $f$ is convex, the sufficient condition above would mean such a stationary point is a global minimizer, and thus a solution exists. I'm unsure how to proceed at this point, and I feel like I'm missing something elementary.  For (ii) above, I believe that I would need to show that $f$ is strictly convex. If $f$ is strictly convex, then the minimizer of $$ \min_{\mathbf{x}\in\mathcal{C}}{f(\mathcal{x})} $$ is unique.  I would greatly appreciate any hints as to how I could proceed with these problems, particularly (i).
Suppose that C is a closed convex subset of $\mathbb R^n$ and $x \in \mathbb R^n$. The projection of $\mathbf x$ onto C is the closest point $\mathbf y \in C : \mathbf z = \mathbf y$ minimizes ||$\mathbf z - \mathbf x$||$_2$ over all $\mathbf z \in C$.  (a) Show that the minimum exists.  (b) Show that there is only one minimizer: that is, show that if $\mathbf y_1$ and $\mathbf y_2$ both minimize ||$\mathbf x - \mathbf z$|| then $\mathbf y_1 = \mathbf y_2$   My ideas:  For (a), to show that the minimum exists I will need to show that it is closed and bounded. We already know it's closed because C is a closed convex subset so I just need to show it's bounded. Am I wanting to minimize ||$\mathbf x - \mathbf z$||$_2$? This is just the Euclidean norm which would be $\sqrt{(\mathbf z - \mathbf x)^2}$=$\sqrt{\mathbf z^2 -\mathbf z \mathbf x -\mathbf x \mathbf z + \mathbf x^2}$. I'm not sure where to go from here...  For (b) I want to use proof by contradiction- so I will assume that $\mathbf y_1$ and  $\mathbf y_2$ both minimize the function and then I'll somehow see that they are in fact equal. How do I set up this argument?
What happened to Finn's blaster at Maz Kanata's? When entering, we see that Finn has the blaster that was given to him earlier:    However, when the fighting begins, Finn tells Maz that he needs a weapon and no longer has his blaster with him.  It seems unlikely to have been lost in the first shots, given how it is secured across his body.  Also, Han and Chewie didn't lose their weapons and they are arguably less secure.  Now, when the team gets to the Resistance base, Finn again has his blaster:    So where was this blaster during the battle? When I create a folder or copy files, after the operation finishes I have to manually refresh the Explorer window (F5). For example, when I create a folder, I don't see the "New Folder" until I hit F5.  Does anyone have any idea how to fix this? What is the difference between '/' and '//' when used for division? How to host a single website on multiple servers? I'm planning on releasing an iOS app that downloads video from my server. If hundreds of people on the launch day try to do this, then my website and the services that my server provides will clearly go down. How do you host a website on multiple servers? When someone tries to access my site, how is their request passed to an available server? How do Apple and Google do it? Thanks for your help. Equal and opposite reaction - forces coming in pairs - or just common sense? How to print original variable's name in Python after it was returned from a function? What does "yield break;" do in C#? I have seen this syntax in MSDN: , but I don't know what it does.  Does anyone know? Recovering broken or deleted NTFS partitions Why disable Fast Boot on Windows 8 when having dual booting? Number of solutions to the congruence $x^q \equiv 1 \mod p$. Let $p, q$ be distinct odd primes, I would like to compute the number of solution $\mod p$ to the congruence $x^q \equiv 1 \mod p$. 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When I got my DSLR, I took a lot of pics and all of them came excellent, even though I had VR at the "Off" position.   Since those came out so well, I wonder when should I turn VR on, and when I should just leave it off.   I found  which answer this question in detail, but that's too much for me to understand easily. Can someone explain the strict rules in short?  Should I use VR with tripod? When should I use it when I'm not using a tripod, and when should I leave it off? Suppose I have a node in the shape of a circle. How can I make its border thicker?   \node[circle, draw=blue!80, inner sep=0pt, minimum size=12pt] (1) at (0,0) {1}; How can I manually generate a random number from a given distribution, as for instance, 10 realisations from the standard normal distribution? To hyphenate or not? As a non-native speaker of English and an engineer by training, I always get confused about hyphenation and almost always end up referring to Google every time I need to make that decision.  Does anybody know of a concise, comprehensive style guide to hyphenation that explains this rather complex issue, once and for all? What is recursion and when should I use it? One of the topics that seems to come up regularly on mailing lists and online discussions is the merits (or lack thereof) of doing a Computer Science Degree. An argument that seems to come up time and again for the negative party is that they have been coding for some number of years and they have never used recursion.  So the question is:   What is recursion? When would I use recursion? Why don't people use recursion? I am working on a print for my wall and I am trying to create something like this:      I have already created the letters in Illustrator and now I am adding the colors in Illustrator and then exporting everything to Photoshop and then using the liquify tool to blend the colors to create a similar effect. I also used the healing patch tool to blend in the hard edges.  Is that the right way of doing it? Or is there another way? "Welcome to emergency mode!" Think it is a fsck problem My computer booted to a black screen with this error message.  Welcome to emergency mode! After logging in,type "journalctl -xb" to view    system logs, "systemctl reboot" to reboot, "systemctl default" or ^D to    try again to boot into default mode.   journalctl -xb snippet (what I think is wrong):  -- Unit systemd-fsckd.service has begun starting up. juli 09 15:40:16 kim-SSD-Sationary systemd-fsck[414]: /dev/sdb1 contains a file system with errors, check forced. juli 09 15:40:16 kim-SSD-Sationary systemd-fsck[414]: /dev/sdb1: Inodes that were part of a corrupted orphan linked list found. juli 09 15:40:16 kim-SSD-Sationary systemd-fsck[414]: /dev/sdb1: UNEXPECTED INCONSISTENCY; RUN fsck MANUALLY. juli 09 15:40:16 kim-SSD-Sationary systemd-fsck[414]: (i.e., without -a or -p options) juli 09 15:40:16 kim-SSD-Sationary systemd-fsck[414]: fsck failed with error code 4. juli 09 15:40:16 kim-SSD-Sationary systemd-fsck[414]: Running request emergency.target/start/replace juli 09 15:40:16 kim-SSD-Sationary systemd[1]: systemd-fsck-root.service: main process exited, code=exited, status=1/FAILURE juli 09 15:40:16 kim-SSD-Sationary systemd[1]: Failed to start File System Check on Root Device. -- Subject: Unit systemd-fsck-root.service has failed -- Defined-By: systemd -- Support: http://lists.freedesktop.org/mailman/listinfo/systemd-devel --  -- Unit systemd-fsck-root.service has failed. --  -- The result is failed. juli 09 15:40:16 kim-SSD-Sationary systemd[1]: Unit systemd-fsck-root.service entered failed state. juli 09 15:40:16 kim-SSD-Sationary systemd[1]: systemd-fsck-root.service failed. juli 09 15:40:16 kim-SSD-Sationary systemd[1]: Starting Remount Root and Kernel File Systems... -- Subject: Unit systemd-remount-fs.service has begun start-up -- Defined-By: systemd -- Support: http://lists.freedesktop.org/mailman/listinfo/systemd-devel   I ignored other errors like: ACPI PCC probe failed, nvidia not proprietary, etc.  I can access my PC by pressing Ctrl+D , but it is annoying. How to use GCC 4.7 in place of 4.6? So I'm working on a project and I wanted to use some C11 features for C. To use those features GCC 4.7 is required, so I went to synaptic and looked up 4.7, and I found the packages gcc-4.7-base so I installed it[1].   So now that gcc-4.7-base is installed I expected to be able to use 4.7, but no, when I do gcc --version it still shows 4.6.3 even more, when I try to directly call 4.7(via /usr/bin/gcc-4.7) it doesn't exist, though a gcc-4.6 executable does exist.  How do I "activate" GCC 4.7? I read , you do some crazy stuff with update-alternatives, but it doesn't work for me as the gcc-4.7 executable doesn't actually exist on my machine.   Notes: 1. I also saw gcc-4.6-base, so I figured I'd uninstall it, but synaptic attempted to remove every package I had installed and install a Java runtime, so for obvious reason I decided not to remove it. "have" vs."have got" in American and British English