Steiner point

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The centre of the mass, distributed over the surface of a convex body, with density equal to the Gaussian curvature. For a non-smooth body it is defined by mixed volumes (see Mixed-volume theory). The Steiner point is additive with respect to the addition of bodies. The centre of a mass, distributed over a contour of variable curvature in the plane, was first studied by J. Steiner in 1840.


[1] B. Gruenbaum, "Measures of symmetry for convex sets" V.L. Klee (ed.) , Convexity , Proc. Symp. Pure Math. , 7 , Amer. Math. Soc. (1963) pp. 238–270
[2] R. Schneider, "Krümmungsschwerpunkte konvexer Körper (I)" Abh. Math. Sem. Univ. Hamburg , 37 : 1–4 (1972) pp. 112–132


Axiomatic characterizations of the Steiner point are treated in [a1], [a2].


[a1] R. Schneider, "On Steiner points of convex bodies" Israel J. Math. , 9 (1971) pp. 241–249
[a2] E.D. Positsel'skii, "Characterization of Steiner points" Math. Notes , 14 (1973) pp. 698–700 Mat. Zametki , 14 (1973) pp. 243–247
[a3] B. Grünbaum, "Convex polytopes" , Wiley (1967)
How to Cite This Entry:
Steiner point. V.A. Zalgaller (originator), Encyclopedia of Mathematics. URL:
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098