Helly's theorem

On the plane, a nonempty intersection of all triples of convex figures implies that the intersection of all nonempty

Helly's theorem is a classical result of combinatorial geometry and convex analysis . The theorem gives a condition on a family of convex sets, guaranteeing that this family has a nonempty intersection.

Content

Wording

End families

Let's pretend that

X_{1},X_{2},\dots ,X_{n}

{\ displaystyle X_ {1}, X_ {2}, \ dots, X_ {n}}

there is a finite family of convex subsets of Euclidean space $\mathbb {R} ^{d}$ ${\ displaystyle \ mathbb {R} ^ {d}}$ such that the intersection of any $d+1$ ${\ displaystyle d + 1}$ of which is nonempty.

Then the intersection of all subsets of this family is nonempty, i.e.

\bigcap _{j=1}^{n}X_{j}\neq \emptyset

{\ displaystyle \ bigcap _ {j = 1} ^ {n} X_ {j} \ neq \ emptyset}

.

Endless Families

For infinite families, it is necessary to additionally require compactness:

Let be $\{X_{\alpha }\}$ ${\ displaystyle \ {X _ {\ alpha} \}}$ there is an arbitrary family of convex compact subsets $\mathbb {R} ^{d}$ ${\ displaystyle \ mathbb {R} ^ {d}}$ such that the intersection of any $d+1$ ${\ displaystyle d + 1}$ of which is nonempty. Then the intersection of all subsets of this family is nonempty.

Consequences

Jung's Theorem: Let $S$ ${\ displaystyle S}$ there are a finite set of points in $d$ ${\ displaystyle d}$ -dimensional Euclidean space $\mathbb {R} ^{d}$ ${\ displaystyle \ mathbb {R} ^ {d}}$ such that any $d+1$ ${\ displaystyle d + 1}$ points from $S$ ${\ displaystyle S}$ can be covered with a single ball. Then everything is a lot $S$ ${\ displaystyle S}$ can be covered with a single ball.
Kirshbraun's theorem

Variations and generalizations

Let be $H$ ${\ displaystyle H}$ - Hilbert space (not necessarily separable ) and $X_{\alpha }$ ${\ displaystyle X _ {\ alpha}}$ - a family of closed bounded convex subsets $H$ ${\ displaystyle H}$ . If the intersection of an arbitrary finite subfamily $X_{\alpha }$ ${\ displaystyle X _ {\ alpha}}$ not empty then $\cap _{\alpha }X_{\alpha }$ ${\ displaystyle \ cap _ {\ alpha} X _ {\ alpha}}$ also not empty.

History

The theorem was proved by Edward Helly in 1913, about which he told Radon , he published it only in 1923 ^[1] , after the publications of Radon ^[2] and Koenig ^[3] .

Notes

↑ E. Helly Über Mengen konvexer Körper mit gemeinschaftlichen Punkten , - Jber. Deutsch. Math. Vereinig. 32 (1923), 175-176.
↑ J. Radon Mengen konvexer Körper, die einen gemeinsamen Punkt enthalten , - Math. Ann. 83 (1921), 113-115.
↑ D. König Über konvexe Körper, - Math. Z. 14 (1922), 208-220.

Literature

Danzer L., Grunbaum B. , Helly's theorem and its applications. - M: Mir, 1968 .-- 159 p.

[1] E. Helly Über Mengen konvexer Körper mit gemeinschaftlichen Punkten , - Jber. Deutsch. Math. Vereinig. 32 (1923), 175-176.

[2] J. Radon Mengen konvexer Körper, die einen gemeinsamen Punkt enthalten , - Math. Ann. 83 (1921), 113-115.

[3] D. König Über konvexe Körper, - Math. Z. 14 (1922), 208-220.