# The first meaningful theorem about convex sets

What is historically first meaningful theorem about general convex sets?

(My guess would be Hyperplane separation theorem, but I can not see any reliable information on who and when it is proved.)

It is not completely clear what "general" means in your question. In arbitrary dimension? Descartes theorem in dimension 3 gives a relation between solid and flat angles of a convex polytope which is equivalent to the Gauss--Bonnet for this case, and implies Euler's theorem. The first results on convex polytopes in arbitrary dimension are probably due to Cauchy and Fourier in connection with linear programming (Cauchy actually had something like simplex method). Also Steiner was working on convex polytopes in the first part of 19s century.

But if "general" really means arbitrary convex sets in any dimension, then it is possible indeed that it is the separation theorem. According to Wikipedia it was proved by Edward Helly in 1912, in an infinite dimensional space but not in full generality. Hahn and Banach proved the general version in 1920. What is usually called "Helly's theorem" in the theory of convex sets in finite dimension was proved in 1913.

EDIT. Geometrie der Zahlen of Minkowski is 1896 (vol. I) ans 1910 (vo. II).
Search in the Jahrbuch for "konvex" in the title gives only one 19 century paper, and it is about convex curves, not general convex sets.

• For me 2-dimensional case is general enough. Wikipedia says that Hyperplane separation theorem was proved by Minkowski, if this is true, it is likely before Helly's theorem, but earliest end of 19th century. So can one say that convexity was not considered until end of 19th century? – Anton Petrunin Dec 16 '15 at 17:15
• Convexity was considered: Cauchy's famous rigidity theorem, for example. I just was not sure about the meaning of the word "general" in your message. – Alexandre Eremenko Dec 16 '15 at 17:29
• Or do you mean by "general" more general than polygons/polytopes? – Alexandre Eremenko Dec 16 '15 at 17:46
• Convexity is a circle of ideas, and Cauchy's theorem is not exactly from there. – Anton Petrunin Dec 16 '15 at 18:01
• OK. Among the early explorers of this circles of ideas I mentioned Fourier, Cauchy (linear programing) and Steiner, all in the early 19 century. – Alexandre Eremenko Dec 16 '15 at 23:25

Convex set: The German term appears in E. Steinitz, "Bedingt konvergente Reihen und konvexe Systeme, I," J. Reine Angew. Math., 143, (1913) 128-175

so perhaps that is a place to look for a "meaningful theorem". Then there would remain the question of such meaningful theorems about convex sets from before the terminology "convex set" was coined.

• This is certainly not the earliest use of the word. – Alexandre Eremenko Dec 16 '15 at 17:52
• @AlexandreEremenko... Great, we can correct the web page then. Of course it has earlier references for "convex", "convex polygon", and "convex function". Just not "convex set". – Gerald Edgar Dec 16 '15 at 18:08
• Now I understand. But this is another matter. The problem is that the word "set" was not used in its present meaning until the early 20 century. The 19 century papers that I mean use convex curves, surfaces, polytopes, functions etc. There was no such thing as "set" :-) – Alexandre Eremenko Dec 16 '15 at 23:21