Let $K$ be a closed convex cone in an n-dimensional Euclidean space. Suppose $K$ has non-empty interior. For $t > 0$ form the subcone $K_t$ consisting of all points in $K$ which lie a distance $t$ or greater from the boundary of $K$.

**True or False?** $K_t$ is a translate of $K$.

In other words, can we obtain $K_t$ by pushing $K$ into its interior?

*Motivation:* this problem arose while trying to establish
the ‘optimal worst-case escape rate’ from the collision locus
in the $N$-body problem. In the collinear equal mass $N$-body problem, the $N-1$ dimensional cone $K$ of interest is the one
defined by the inequalities $ x_1 \le x_2 \le \dotsb \le x_N$
within the hyperplane $\sum x_i = 0$ of ${\mathbb R}^N$.
(You might call this the “$A_N$ cone” since it is a Weyl chamber for the $A_N$ Lie algebra.)

*What do I know?*

Not much beyond two dimensions and a lot of special cases. In two dimensions the assertion is true. $K$ is a sector. $K_t$ is the translate of $K$ by a vector directed along the bisector of the angle formed by the sector's boundary. It follows that the assertion also holds for three-dimensional trihedral cones: i.e. those $K$'s in 3-space whose affine cross-section
is a triangle. It seems a dissection + limit argument ought
lead from here to the general 3-dimensional result but I did not get even this.

*Convex geometry terminology?* If the assertion is true, then the length of the translation vector taking $K$ to $K_1$ measures of the “sharpness” of the cone $K$. This length is the reciprocal of $m(K)= \max_{q \in \operatorname{int}(K)} \operatorname{dist}(q, \partial K)/\|q\|$ as $q$ varies over the interior of $K$. Does any one know the convex geometry term for this $m(K)$ or its reciprocal, the “sharpness” of $K$?

`\sum`

, not`\Sigma`

(compare $\Sigma x_i = 0$`\Sigma x_i = 0`

to $\sum x_i = 0$`\sum x_i = 0`

). MathJax note: backticks are parsed by the MathJax parser, not by TeX, so having too many of them in a paragraph can result in the intervening text being effectively wrapped in`<pre>`

, as happened in this post. It's better to use`" "`

or`“ ”`

on MO. I edited accordingly for both. $\endgroup$