<p>Suppose we have two convex bodies $A$ and $B$, where $A \subseteq B$.  Is it always true that $\mathrm{Vol}(A)/\mathrm{SurfaceArea}(A) \leq \mathrm{Vol}(B)/\mathrm{SurfaceArea}(B)$?</p>

<p>It's true in all the examples I've tried, but I'm not sure how to prove the general case, or whether the general case is even true.</p>

CC BY-SA 3.0
2016-08-03T07:46:45.017Z
This question is probably better suited for http://mathoverflow.net/ or http://math.stackexchange.com/
Jan Johannsen
CC BY-SA 3.0
2016-08-03T08:21:03.120Z
A note: the isoperimetric constant is usually defined in a scale-invariant way, i.e. $S^{1/n-1}/V^{1/n}$, where $S$ is surface area (Hausdorff measure or Minkowski content), $V$ is volume (Lebesgue measure) and $n$ is the dimension.
Sasho Nikolov
CC BY-SA 3.0
2016-08-05T14:00:25.337Z
cross-posted at http://math.stackexchange.com/questions/1877183/do-nested-convex-bodies-have-nested-isoperimetric-ratios
Sasho Nikolov
CC BY-SA 3.0
2016-08-05T16:02:50.747Z
Thanks, but the lack of scale invariance is intentional.  (The corresponding scale invariant inequality would be false.)
Maximinus
CC BY-SA 3.0
2016-08-05T17:21:23.757Z
Please [do not post the same question on multiple sites](http://meta.stackexchange.com/q/64068). Each community should have an honest shot at answering without anybody's time being wasted.
D.W.
<convex-optimization><convex-geometry><high-dimensional-geometry>
Do nested convex bodies have increasing "Volume/Surface Area" ratios?
Maximinus
# Do nested convex bodies have increasing "Volume/Surface Area" ratios?

(Asked by: username_0 on 2016-08-02)

Suppose we have two convex bodies $A$ and $B$, where $A \subseteq B$. Is it always true that $\mathrm{Vol}(A)/\mathrm{SurfaceArea}(A) \leq \mathrm{Vol}(B)/\mathrm{SurfaceArea}(B)$?

It's true in all the examples I've tried, but I'm not sure how to prove the general case, or whether the general case is even true.

**Tags:** convex-optimization, convex-geometry, high-dimensional-geometry

> **Comments:**
> **username_1**: This question is probably better suited for http://mathoverflow.net/ or http://math.stackexchange.com/  
> **username_2**: A note: the isoperimetric constant is usually defined in a scale-invariant way, i.e. $S^{1/n-1}/V^{1/n}$, where $S$ is surface area (Hausdorff measure or Minkowski content), $V$ is volume (Lebesgue measure) and $n$ is the dimension.  
> **username_2**: cross-posted at http://math.stackexchange.com/questions/1877183/do-nested-convex-bodies-have-nested-isoperimetric-ratios  
> **username_0**: Thanks, but the lack of scale invariance is intentional.  (The corresponding scale invariant inequality would be false.)  
> **username_3**: Please [do not post the same question on multiple sites](http://meta.stackexchange.com/q/64068). Each community should have an honest shot at answering without anybody's time being wasted.  

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