TITLE: Regularity of convex sets in $\mathbb{R}^n$
QUESTION [11 upvotes]: 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 so it could be that his book is the first place where it was proved. However, the statement seems so classical that I tend to believe the result had been known before.
It is very likely that I will refer to this result in my research and I would like to give a proper reference to it.
Edit. It seems that there are earlier references than the one in the answer, see the comments below the answer. That means the questions might still have a better answer.
[1] L. Hörmander,
Notions of convexity.
Progress in Mathematics, 127. Birkhäuser Boston, Inc., Boston, MA, 1994.

REPLY [5 votes]: To my knowledge, this is the first place where I saw the result being claimed:
C. O. Kiselman. Regularity classes for operations in convexity theory. Kodai Math. J. 15. 1992.
In particular on the first page the author states:

To  describe   the  simplest   case   of   our  results,   let  $A$   be  a  compact  set  in  $R^n$. If  the  boundary  of  $A$   is   of   class   $C^{1,1}$,  then  $A$   is  a  union  of   Euclidean  balls   with radii   bounded   from   below;   if   $A$   is  convex,   the  converse   holds.