# Do nested convex bodies have increasing “Volume/Surface Area” ratios? [closed]

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.

• This question is probably better suited for mathoverflow.net or math.stackexchange.com – Jan Johannsen Aug 3 '16 at 7:46
• 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 Aug 3 '16 at 8:21
• cross-posted at math.stackexchange.com/questions/1877183/… – Sasho Nikolov Aug 5 '16 at 14:00
• Thanks, but the lack of scale invariance is intentional. (The corresponding scale invariant inequality would be false.) – Maximinus Aug 5 '16 at 16:02
• Please do not post the same question on multiple sites. Each community should have an honest shot at answering without anybody's time being wasted. – D.W. Aug 5 '16 at 17:21