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Let $P\subset\Bbb R^d$ be a convex polytope (convex hull of finitely many points). A $k$-in-sphere of $P$ is a sphere centered at the origin to which each $k$-face of $P$ is tangent to. So a 0-in-sphere contains all the vertices and is actually a circumsphere, and a $(d-1)$-in-sphere is completely contained in $P$.

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Question: If $P$ has $k$-in-spheres for all $k\in\{0,...,d-1\}$, is $P$ a regular polytope?

By definition, all these spheres are centered at the origin, hence are concentric.

The answer to the question is Yes for polygons. For $d\ge 3$ note that this property of $P$ is inherited by its faces, and it follows that all 2-faces of $P$ are regular polygons and all edges are of the same length.

Let $P\subset\Bbb R^d$ be a convex polytope (convex hull of finitely many points). A $k$-in-sphere of $P$ is a sphere centered at the origin to which each $k$-face of $P$ is tangent. So a 0-in-sphere contains all the vertices and is actually a circumsphere, and a $(d-1)$-in-sphere is completely contained in $P$.

$\qquad\qquad\qquad\qquad\qquad$

Question: If $P$ has $k$-in-spheres for all $k\in\{0,...,d-1\}$, is $P$ a regular polytope?

By definition, all these spheres are centered at the origin, hence are concentric.

The answer to the question is Yes for polygons. For $d\ge 3$ note that this property of $P$ is inherited by its faces, and it follows that all 2-faces of $P$ are regular polygons and all edges are of the same length.

Let $P\subset\Bbb R^d$ be a convex polytope (convex hull of finitely many points). A $k$-in-sphere of $P$ is a sphere around the origin to which each $k$-face of $P$ is tangent to. So a 0-in-sphere contains all the vertices and is actually a circumsphere, and a $(d-1)$-in-sphere is completely contained in $P$.

$\qquad\qquad\qquad\qquad\qquad$

Question: If $P$ has $k$-in-spheres for all $k\in\{0,...,d-1\}$, is $P$ a regular polytope?

By definition, these spheres are required to be concentric.

The answer to the question is Yes for polygons. For $d\ge 3$ note that this property of $P$ is inherited by its faces, and it follows that all 2-faces of $P$ are regular polygons and all edges are of the same length.

Let $P\subset\Bbb R^d$ be a convex polytope (convex hull of finitely many points). A $k$-in-sphere of $P$ is a sphere centered at the origin to which each $k$-face of $P$ is tangent to. So a 0-in-sphere contains all the vertices and is actually a circumsphere, and a $(d-1)$-in-sphere is completely contained in $P$.

$\qquad\qquad\qquad\qquad\qquad$

Question: If $P$ has $k$-in-spheres for all $k\in\{0,...,d-1\}$, is $P$ a regular polytope?

By definition, all these spheres are centered at the origin, hence are concentric.

The answer to the question is Yes for polygons. For $d\ge 3$ note that this property of $P$ is inherited by its faces, and it follows that all 2-faces of $P$ are regular polygons and all edges are of the same length.

Let $P\subset\Bbb R^d$ be a convex polytope (convex hull of finitely many points). A $k$-in-sphere of $P$ is a sphere around the origin to which each $k$-face of $P$ is tangent to. So a 0-in-sphere contains all the vertices and is actually a circumsphere, and a $(d-1)$-in-sphere is completely contained in $P$.

$\qquad\qquad\qquad\qquad\qquad$

Question: If $P$ has $k$-in-spheres for all $k\in\{0,...,d-1\}$, is $P$ a regular polytope?

Note that the spheres are concentric. The answer is Yes for polygons. For $d\ge 3$ note that this property of $P$ is inherited by its faces, and it follows that all 2-faces of $P$ are regular polygons and all edges are of the same length.

Let $P\subset\Bbb R^d$ be a convex polytope (convex hull of finitely many points). A $k$-in-sphere of $P$ is a sphere around the origin to which each $k$-face of $P$ is tangent to. So a 0-in-sphere contains all the vertices and is actually a circumsphere, and a $(d-1)$-in-sphere is completely contained in $P$.

$\qquad\qquad\qquad\qquad\qquad$

Question: If $P$ has $k$-in-spheres for all $k\in\{0,...,d-1\}$, is $P$ a regular polytope?

By definition, these spheres are required to be concentric. 

The answer to the question is Yes for polygons. For $d\ge 3$ note that this property of $P$ is inherited by its faces, and it follows that all 2-faces of $P$ are regular polygons and all edges are of the same length.