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.

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    $\begingroup$ This question is probably better suited for mathoverflow.net or math.stackexchange.com $\endgroup$ – Jan Johannsen Aug 3 '16 at 7:46
  • $\begingroup$ 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. $\endgroup$ – Sasho Nikolov Aug 3 '16 at 8:21
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    $\begingroup$ cross-posted at math.stackexchange.com/questions/1877183/… $\endgroup$ – Sasho Nikolov Aug 5 '16 at 14:00
  • $\begingroup$ Thanks, but the lack of scale invariance is intentional. (The corresponding scale invariant inequality would be false.) $\endgroup$ – Maximinus Aug 5 '16 at 16:02
  • $\begingroup$ 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. $\endgroup$ – D.W. Aug 5 '16 at 17:21