<p>I've begun reading the paper &quot;Random Walks in a Convex Body and an Improved Volume Algorithm&quot; by Lovász-Simonovits ('93). Although the paper for the most part is pretty self-contained and easy to read, the proof of the Localization lemma seems to be rather obtuse to me. Importantly there are certain statements that I'm having a hard part understanding. I would love to know any good exposition and/or if someone could help me out with these questions.</p>
<ul>
<li><p>Under Claim 2, there is this following statement: It follows by a standard compactness arguments that such a pair of points and
function <span class="math-container">$\psi$</span> exists. I am having a hard time understanding this argument.</p>
</li>
<li><p>I'm having a hard time understanding what the convex body <span class="math-container">$K^\prime$</span> actually represents. You can give me an example in the case of 2-D or 3-D convex bodies.</p>
</li>
<li><p>Can anyone explain what the body <span class="math-container">$L_A$</span> exactly represents?</p>
</li>
</ul>
<p>Really appreciate any help that I can get</p>

CC BY-SA 4.0
2023-10-09T12:55:13.783Z
link? ${}{}{}{}{}$
mathworker21
CC BY-SA 4.0
2023-10-09T14:39:00.880Z
@mathworker21 http://faculty.washington.edu/harin/LS1.pdf
total dependent random choice
CC BY-SA 4.0
2023-10-09T18:18:14.297Z
https://cs.stackexchange.com/q/162383/755
D.W.
<convex-geometry><random-walks><convex-hull>
Detailed exposition for proof of Localization Lemma in paper "Random Walks in a Convex Body and an Improved Volume Algorithm"
total dependent random choice
# Detailed exposition for proof of Localization Lemma in paper "Random Walks in a Convex Body and an Improved Volume Algorithm"

(Asked by: username_0 on 2023-10-09)

I've begun reading the paper "Random Walks in a Convex Body and an Improved Volume Algorithm" by Lovász-Simonovits ('93). Although the paper for the most part is pretty self-contained and easy to read, the proof of the Localization lemma seems to be rather obtuse to me. Importantly there are certain statements that I'm having a hard part understanding. I would love to know any good exposition and/or if someone could help me out with these questions.

* Under Claim 2, there is this following statement: It follows by a standard compactness arguments that such a pair of points and
  function $\psi$ exists. I am having a hard time understanding this argument.
* I'm having a hard time understanding what the convex body $K^\prime$ actually represents. You can give me an example in the case of 2-D or 3-D convex bodies.
* Can anyone explain what the body $L\_A$ exactly represents?

Really appreciate any help that I can get

**Tags:** convex-geometry, random-walks, convex-hull
> **Comments:**
> **username_1**: link? ${}{}{}{}{}$  
> **username_0**: @mathworker21 http://faculty.washington.edu/harin/LS1.pdf  
> **username_2**: https://cs.stackexchange.com/q/162383/755  

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## 0 answers

