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Let's say I have a convex polytope $\mathcal{P} \subset \mathbb{R}^n$ with non-empty interior.

Let $\mathcal{P}=\bigcap_{i=1}^mH_i$ for some halfspaces $H_i$ and let $d=diam(\mathcal{P})=\sup\{\|x-y\|:x,y\in \mathcal{P}\}$ be the diameter of the polytope.

If I define a new polytope $\mathcal{P}_{\lambda}$ by translating all facets (the $n-1$ dimensional faces) of $\mathcal{P}$ a distance $\lambda$ in their inward normal direction, how does the diameter $d_\lambda=diam(\mathcal{P}_\lambda)$ relate to the diameter $d$ of the original polytope $\mathcal{P}$, the parameter $\lambda$, and the dimension $n$ (if it depends on $n$)?

We can assume that $\lambda$ is small enough so that $\mathcal{P}_\lambda$ has no empty interior.

I have found that $\mathcal{P}_\lambda$ is called an "inner parallel body" of $\mathcal{P}$.

Anyone got an idea?

EDIT: I found that the inradius $r_\lambda$ of $\mathcal{P}_\lambda$ (that is the radius of the maximum ball inside $\mathcal{P}_\lambda$) is $r_\lambda=r-\lambda$ where $r$ is the inradius of the original polytope $\mathcal{P}$.

Is a similar relation true for the circumradius (i.e the radius of the minimal ball containing the polytope)?

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    $\begingroup$ In the first sentence of the question, $n$ is the dimension of the ambient space $\mathbb R^n$. In the second sentence, it's the number of half-spaces $H_i$. Those should surely be different. $\endgroup$
    – Andreas Blass
    Commented Mar 8, 2022 at 23:44
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    $\begingroup$ Couldn't $d_{\lambda}$ be arbitrarily small? If you think about a really skinny rhombus in the plane, then $\frac{d}{d\lambda} d_\lambda |_{\lambda = 0}$ can be made arbitrarily large. $\endgroup$
    – Willie Wong
    Commented Mar 9, 2022 at 1:55
  • $\begingroup$ @AndreasBlass Yes thanks, I changed it now. $\endgroup$
    – Jjj
    Commented Mar 9, 2022 at 10:07
  • $\begingroup$ @WillieWong I want to know how $d_\lambda$ changes with $\lambda$ and the original diameter $d$. $\endgroup$
    – Jjj
    Commented Mar 9, 2022 at 10:12
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    $\begingroup$ @jonatan: my point is that you cannot just write $d_{\lambda}$ as a function of $d$ and $\lambda$. You need more information about the original polytope. $\endgroup$
    – Willie Wong
    Commented Mar 9, 2022 at 18:19

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