User ak47 - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T20:12:12Z http://mathoverflow.net/feeds/user/29366 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/122709/some-basics-of-prekopa-leindler-inequality some basics of Prekopa-Leindler inequality ak47 2013-02-23T11:02:04Z 2013-02-23T11:02:04Z <p>I'm familiar with Prekopa-Leindler as stated in <a href="http://en.wikipedia.org/wiki/Pr%C3%A9kopa%E2%80%93Leindler_inequality" rel="nofollow">http://en.wikipedia.org/wiki/Pr%C3%A9kopa%E2%80%93Leindler_inequality</a>, for example. </p> <p>can one say, given $h,f,g$ that satisfy $\forall x,y ,$ $h(tx+(1-t)y) \geq f(x) ^{t} g(y)^{1-t}$ that $$\int e^{h(z)}dz \geq (\int e^{f(x)}dx)^{t} (\int e^{g(y)}dy)^{1-t}$$</p> http://mathoverflow.net/questions/122706/regarding-displacement-convexity-and-perkopa-leindler Regarding displacement convexity and Perkopa-Leindler ak47 2013-02-23T10:54:13Z 2013-02-23T10:54:13Z <p>Hi, I have some questions regarding this subject. The main reference I used is the paper "Displacement convexity of entropy and related inequalities on graphs" / nathael gozlan, cyril roberto, paul-marie samson, prasad tetali.</p> <p>1) for a meaure $\mu$ on {0,1}, what is the formal definition of the n-fold product $\mu^{\otimes n}$?</p> <p>Some background before continuing (need to take a big breath) Theorem 6.4 in this paper states the discrete version of Perkopa Leindler: Let n ∈ N$^*$, t ∈ [0, 1] and $\mu \in P(V^n)$. Suppose that µ verifies the following property: for any $v_0, v_1 \in P(V^n)$, there exists a coupling $\pi \in \Pi(v_0,v_1)$ such that $$H(v_t^\pi |\mu) \leq (1-t)H(v_0|\mu) + tH(v_1|\mu) - ct(1-t)I_2^{(n)}(\pi)$$ If $( f , g, h)$ is a triple of functions on $V^n$ such that: $\forall x \in V^n , \forall m \in P(V^n),$<br> $$(*) \int \int h(z)v_t^{x,y}(dz) m(dy) \geq (1-t)f(x)+t \int g(y) m(dy)-c t(1-t)\sum_{i=1}^n (\int d(x_i,y_i)m(dy))^2$$ then it holds $$(**)<br> \int e^{h(z)}\mu(dz) \geq (\int e^{f(x)}\mu(dx))^{1-t} (\int e^{g(y)}\mu(dy))^t<br>$$</p> <p>and throrem 6.7 states the "Perkopa-Leindler for discrete hypercube": Let $\mu$ be a probability measure on {0,1}, $n \in N^{*}$. for all triple $(f,g,h)$ verifying (**) with c=1/2, it holds $$\int e^{h(z)}\mu^{\otimes n}(dz) \geq (\int e^{f(x)} \mu^{\otimes n}(dx))^{1-t} (\int e^{g(y)} \mu^{\otimes n}(dy))^{t}$$</p> <p>so now I can ask the following questions.</p> <p>2) why do we need c = 1/2 in order to deduce theorem 6.7?</p> <p>3) is (*) always true because of displacement convexity property?</p> http://mathoverflow.net/questions/114325/convexity-in-0-1-n Convexity in $\{0,1\}^n$ ak47 2012-11-24T11:19:48Z 2013-02-15T16:25:07Z <p>how is convexity defined in a subset $A \subset \{0,1\}^n$? furthermore, is there any extention of the Brunn-Minkowski inequality for subsets of $\{0,1\}^n$? thanks.</p> <p><strong>Edit</strong> (previously posted as an answer) thank you for the reference article of Ollivier and Villani. I have a few misunderstanding though, regarding Brunn-Minkowski in $R^n$ and {0,1}$^n$(the hyperplane).</p> <ol> <li><p><strong>Edit - this is the corrected question:</strong> In $R^n$, there is a claim that it is enough to show that $$|\lambda A+(1-\lambda)B| \geq |A|^{\lambda}|B|^{1-\lambda}, \forall 0 \leq \lambda \leq 1$$ in order to conclude BM (Brunn-Minkowski) inequality: $$|A+B|^{1/n} \geq |A|^{1/n} + |B|^{1/n}$$ However, I couldn't think of how to prove it. Is it a trivial claim? How can someone prove it?</p></li> <li><p>In Ollivier and Villani's paper, it handles $M$, the middle points between a and b in the hypercube. I don't understand how can we expand this theory for $M'=\frac{1}{4}A+\frac{3}{4}B$, for example. we need it, I think, in order to conclude the real BM inequality in hypercube.</p></li> <li><p>I feel there is a basic difference between BM inequality in $R^n$ and in the hypercube: in $R^n$ we claim $|A+B|^{1/n} \geq |A|^{1/n} + |B|^{1/n}$, with 1/n power-factor is quite intuitive since volume of balls in $R^n$ is $\sim r^n$. but balls in hyperplane don't grow that way... so, I assume the formula of the MB inequality should look different: $$\phi(|A+B|) \geq \phi(|A|)+\phi(|B|)$$. Is there any idea of how $\phi$ should look like? </p></li> </ol> http://mathoverflow.net/questions/116369/monotone-sets-and-growth-in-the-hypercube Monotone sets and growth in the Hypercube ak47 2012-12-14T13:01:37Z 2012-12-14T13:01:37Z <ol> <li><p>Is the known definition for a monotone set in the hyperplane is : a set $A \in${$0,1$}$^n$ such that for every $a \in A$ if $b &lt; a$ so $b \in A$?</p></li> <li><p>Given all hamming balls and all boxes (vector subspaces) in {$0,1$}$^n$. Can I construct a genereal set $A$ in the hypercube, as a minkowski sum of balls and boxes, that is defined uniquely?</p></li> <li><p>What is known about the sequence $|A|,|A+B|,|A+2B|,\ldots$ for A a general set in hypercube, and B is the hamming ball of radius 1?</p></li> </ol> http://mathoverflow.net/questions/114325/convexity-in-0-1-n/114335#114335 Answer by ak47 for Convexity in $\{0,1\}^n$ ak47 2012-11-24T14:13:46Z 2012-11-25T20:34:49Z <p><em>(see the question edit)</em></p> http://mathoverflow.net/questions/116369/monotone-sets-and-growth-in-the-hypercube Comment by ak47 ak47 2012-12-15T17:02:24Z 2012-12-15T17:02:24Z ok, got you. it's quite trivial :) http://mathoverflow.net/questions/116369/monotone-sets-and-growth-in-the-hypercube Comment by ak47 ak47 2012-12-15T15:04:08Z 2012-12-15T15:04:08Z Can you please write the definitions of $2k$-discrete sets and singleton sets (or reference me to these definitions)? thanks http://mathoverflow.net/questions/114325/convexity-in-0-1-n Comment by ak47 ak47 2012-11-25T22:07:33Z 2012-11-25T22:07:33Z thanks. please see my edited first question. http://mathoverflow.net/questions/114325/convexity-in-0-1-n/114329#114329 Comment by ak47 ak47 2012-11-24T14:14:25Z 2012-11-24T14:14:25Z thank you, my comment is below.