<p>Is it appropriate/meaningful to write vector/points <span class="math-container">$(a,b) \le (c,d)$</span>, where i would mean component wise each component is <span class="math-container">$\le$</span></p>
<ul>
<li>Specifically is my example below with reference to concavity meaningful.</li>
<li>I ask because i know we can start to use <span class="math-container">$\succeq$</span> and be more specific about preference relations, invoking the notions of  non-satiation (monotonicity) etc.</li>
</ul>
<p>For example in the context of wanting to demonstrate that a concave function implies a convex hyper graph, something like this...</p>
<p><span class="math-container">$[λx_1 + (1- λ)x_2, λy_1 + (1- λ)y_2] \le [λx_1 + (1- λ)x_2, f(λx_1 + (1-λ)x_2)]$</span></p>
<ul>
<li>Where <span class="math-container">$y \le f(x)$</span> and <span class="math-container">$f$</span> is concave.</li>
</ul>

CC BY-SA 4.0
2023-04-20T08:33:36.133Z
It is meaningful if you explain that is how you use the term (so be explicit about componentwise use).
Jesper Hybel
CC BY-SA 4.0
2023-04-20T09:03:55.617Z
There is a (small) ambiguity about ≥, because there is ≧. When they are distinguished, ≥ means "elementwise ≥, but at least one strict". This is rare, but I think, stating how you use ≥ would be helpful.
kitsune
CC BY-SA 4.0
2023-04-20T11:47:33.973Z
This is exactly the kind of thing I was wondering, thanks Kitsune!
CormJack
CC BY-SA 4.0
2023-04-20T16:54:41.753Z
In my experience, this notation is used is some context, for example when speaking of order theory, and I agree that must be specified. But I would avoid to use that notation in general analysis, where it is not  usual, the example about concavity looks a little bit strange to me.
BakerStreet
CC BY-SA 4.0
2023-04-20T20:15:31.410Z
Hmmm I see. Well it's definitely fine to do the proofs without writing in the way I did, so maybe I will stick to the more standard approach. And yes I had a whole question on Order Relations ins Math Stack, but people just aren't as helpful there hahah
CormJack
<consumer-theory><optimization><consumption>
Convex Combination of pairs of points
CormJack
# Convex Combination of pairs of points

(Asked by: username_0 on 2023-04-20)

Is it appropriate/meaningful to write vector/points $(a,b) \le (c,d)$, where i would mean component wise each component is $\le$

* Specifically is my example below with reference to concavity meaningful.
* I ask because i know we can start to use $\succeq$ and be more specific about preference relations, invoking the notions of non-satiation (monotonicity) etc.

For example in the context of wanting to demonstrate that a concave function implies a convex hyper graph, something like this...

$[λx\_1 + (1- λ)x\_2, λy\_1 + (1- λ)y\_2] \le [λx\_1 + (1- λ)x\_2, f(λx\_1 + (1-λ)x\_2)]$

* Where $y \le f(x)$ and $f$ is concave.

**Tags:** consumer-theory, optimization, consumption
> **Comments:**
> **username_1**: It is meaningful if you explain that is how you use the term (so be explicit about componentwise use).  
> **username_2**: There is a (small) ambiguity about ≥, because there is ≧. When they are distinguished, ≥ means "elementwise ≥, but at least one strict". This is rare, but I think, stating how you use ≥ would be helpful.  
> **username_0**: This is exactly the kind of thing I was wondering, thanks Kitsune!  
> **username_3**: In my experience, this notation is used is some context, for example when speaking of order theory, and I agree that must be specified. But I would avoid to use that notation in general analysis, where it is not  usual, the example about concavity looks a little bit strange to me.  
> **username_0**: Hmmm I see. Well it's definitely fine to do the proofs without writing in the way I did, so maybe I will stick to the more standard approach. And yes I had a whole question on Order Relations ins Math Stack, but people just aren't as helpful there hahah  

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## 0 answers

