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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 The two answers below show that the result was proved in an unpublished PhD from 1957, it was mentioned without a proof or reference in a paper by Kiselman and the first published proof I am aware of appears in Hormander's book.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.

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  • $\begingroup$ check this? projecteuclid.org/download/pdf_1/euclid.kmj/1138039654 -- the author claims at least one direction of the above result as one of the results of his paper....and with convexity, also the converse $\endgroup$
    – Suvrit
    Commented Nov 18, 2020 at 16:12
  • $\begingroup$ This result (or better said the non trivial implication) follows from a general regularity result for sets of positive reach, stated for the first time without a proof in Federer´s paper on curvature measures (see Remark 4.20). See also the recent paper of Rataj-Zajicek "On the stucture of sets with positive reach" for a description of these results. $\endgroup$
    – Longyearbyen
    Commented Jun 20, 2021 at 10:06

2 Answers 2

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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.

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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.

Screenshot of Kiselman's paper

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  • $\begingroup$ I only had a quick look at the paper. Kiselman states the result in the introduction, but does he prove it in the paper? I did not see the proof, but I did not spend much time reading it. $\endgroup$
    – Piotr Hajlasz
    Commented Nov 18, 2020 at 16:27
  • $\begingroup$ Oops, I looked at the theorems closely, and it seems that without more careful reading, it's not easy to immediately extract this result (due to a bunch of notation hiding this stuff). But upon reading the paper more closely, it seems that perhaps an even earlier paper should get the credit, though I haven't had a chance to read those references to be certain. $\endgroup$
    – Suvrit
    Commented Nov 19, 2020 at 20:24
  • $\begingroup$ Is that Esperanto in the abstract? $\endgroup$
    – Pedro Lauridsen Ribeiro
    Commented May 27, 2021 at 21:35
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    $\begingroup$ @PedroLauridsenRibeiro Yes, it's Esperanto. (Only the second time I've seen an Esperanto abstract.) $\endgroup$
    – Andreas Blass
    Commented May 28, 2021 at 0:13
  • $\begingroup$ Now I'm wondering if it's common for Japanese math journals to accept texts / abstracts in Esperanto... There is also the journal Funkcialaj Ekvacioj ( = "Functional Equations" in Esperanto, apparently), published by the Department of Mathematics of Kobe University. $\endgroup$
    – Pedro Lauridsen Ribeiro
    Commented May 28, 2021 at 1:56

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