The following result is Proposition 2.4.3 in [1]:



> 
> **Theorem.** Let $K\subset\mathbb{R}^n$ be a bounded convex set with the non-empty interior. Then $\partial K\in C^{1,1}$ if and only if
> there is $r>0$ such that $K$ is the unioin of balls of radius $r$.
> 
> 
> 


**Question.** Do you know who is the author of this result?


Hörmander does not provide any reference.


**Edit.** I am still quite puzzled about the result. The two answers below show that the result was proved in an unpublished PhD from 1957, it was mentioned without a proof or reference in a paper by Kiselman and the first published proof I am aware of appears in Hormander's book. The result is in my opinion very beautiful not entirely trivial so I expect there should be other references.


**I am still waiting for more answers showing other references to published proofs.**


[1] **L. Hörmander**,
*Notions of convexity.*
Progress in Mathematics, 127. Birkhäuser Boston, Inc., Boston, MA, 1994.



Regularity of convex sets in $\mathbb{R}^n$
Theorem 1.3 in:


<https://arxiv.org/pdf/1304.4179.pdf>


and the historical discussion below it is relevant. The result (in a form that applies without the convexity assumption) seems to first appear in 1957.


