<p>Studying concavity for our constrained optimisation problems made me think of the jelly shape...I believe this shape (see Jelly image) isn't quasi-concave and hence isn't concave.  Because it would be possible to have contours such that.</p>
<p><span class="math-container">$f[(1-λ)x_₁ + (λ)x_₂] &gt;$</span> Min{<span class="math-container">$f(x_1), f(x_2)$</span>}</p>
<p>See contour Image:</p>
<ol>
<li>Am I correct?</li>
<li>Is there a clarification for a shape like this, which looks &quot;concave&quot; but isn't by our formal definitions?</li>
</ol>
<p>Thanks for your thoughts!</p>
<p><a href="https://i.sstatic.net/JN41u.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/JN41u.png" alt="enter image description here" /></a>
<a href="https://i.sstatic.net/KMRI7.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/KMRI7.jpg" alt="enter image description here" /></a></p>

CC BY-SA 4.0
2023-02-10T16:52:13.137Z
The jelly shape is, indeed, not the hypograph of a quasi-concave function.
Michael Greinecker
CC BY-SA 4.0
2023-02-14T17:19:58.490Z
Thank you @MichaelGreinecker, if you have any more details to add, on the topic I'd be happy to vote and accept your answer. E.g. is there anything tot he fact that it looks concave, and if you were to "zoom out" far enough so you couldn't see the dimples it would become strictly concave, almost like some kind of limit? Thanks!
CormJack
CC BY-SA 4.0
2023-02-14T18:16:54.337Z
I am not familiar with anything like that. There are ways to measure how nonconvex a set is, such as the Hausdorff distance to convex hull, but I am not sure this is of use to you.
Michael Greinecker
CC BY-SA 4.0
2023-02-14T19:05:14.290Z
Okay thank you @MichaelGreinecker! I appreciate you writing back. Happy to vote and accept those comments if you turn them into an answer...but tbf I don't think you need the points hahah
CormJack
<optimization>
Is the Jelly Shape Quasi-concave?
CormJack
# Is the Jelly Shape Quasi-concave?

(Asked by: username_0 on 2023-02-10)

Studying concavity for our constrained optimisation problems made me think of the jelly shape...I believe this shape (see Jelly image) isn't quasi-concave and hence isn't concave. Because it would be possible to have contours such that.

$f[(1-λ)x\_₁ + (λ)x\_₂] >$ Min{$f(x\_1), f(x\_2)$}

See contour Image:

1. Am I correct?
2. Is there a clarification for a shape like this, which looks "concave" but isn't by our formal definitions?

Thanks for your thoughts!

[![enter image description here](https://i.sstatic.net/JN41u.png)](https://i.sstatic.net/JN41u.png)
[![enter image description here](https://i.sstatic.net/KMRI7.jpg)](https://i.sstatic.net/KMRI7.jpg)

**Tags:** optimization
> **Comments:**
> **username_1**: The jelly shape is, indeed, not the hypograph of a quasi-concave function.  
> **username_0**: Thank you @MichaelGreinecker, if you have any more details to add, on the topic I'd be happy to vote and accept your answer. E.g. is there anything tot he fact that it looks concave, and if you were to "zoom out" far enough so you couldn't see the dimples it would become strictly concave, almost like some kind of limit? Thanks!  
> **username_1**: I am not familiar with anything like that. There are ways to measure how nonconvex a set is, such as the Hausdorff distance to convex hull, but I am not sure this is of use to you.  
> **username_0**: Okay thank you @MichaelGreinecker! I appreciate you writing back. Happy to vote and accept those comments if you turn them into an answer...but tbf I don't think you need the points hahah  

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