Borsuk conjectured that it is possible to cut an -dimensional shape of generalized
diameter 1 into
pieces each with diameter smaller than the original. It is true for , 3 and when the boundary is "smooth." However,
the minimum number of pieces required has been shown to increase as . Since at , the conjecture becomes false at high dimensions.
Kahn and Kalai (1993) found a counterexample in dimension 1326, Nilli (1994) a counterexample in dimension 946. Hinrichs and Richter (2003) showed that the conjecture is false
for all .
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Vollkugel in
Mengen." Verh. Internat. Math.-Kongr. Zürich2, 192, 1932.Borsuk,
K. "Drei Sätze über die -dimensionale euklidische Sphäre." Fund. Math.20,
177-190, 1933.Cipra, B. "If You Can't See It, Don't Believe It...."
Science259, 26-27, 1993.Cipra, B. What's
Happening in the Mathematical Sciences, Vol. 1. Providence, RI: Amer.
Math. Soc., pp. 21-25, 1993.Grünbaum, B. "Borsuk's Problem
and Related Questions." In Convexity: Proceedings of the Seventh Symposium
in Pure Mathematics of the American Mathematical Society, Held at the University
of Washington, Seattle, June 13-15, 1961. Providence, RI: Amer. Math. Soc., pp. 271-284,
1963.Hinrichs, A. and Richter, C. "New Sets with Large Borsuk Numbers."
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"A Counterexample to Borsuk's Conjecture." Bull. Amer. Math. Soc.29,
60-62, 1993.Lyusternik, L. and Schnirel'mann, L. Topological Methods
in Variational Problems. Moscow, 1930.Lyusternik, L. and Schnirel'mann,
L. "Topological Methods in Variational Problems and Their Application to the
Differential Geometry of Surfaces." Uspehi Matem. Nauk (N.S.)2,
166-217, 1947.Nilli, A. "On Borsuk's Problem." Jerusalem
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-dimensional shape of generalized
diameter 1 into 
pieces each with diameter smaller than the original. It is true for 
, 3 and when the boundary is "smooth." However,
the minimum number of pieces required has been shown to increase as 
. Since 
at 
, the conjecture becomes false at high dimensions.
.
-dimensionalen
Vollkugel in 
Mengen." Verh. Internat. Math.-Kongr. Zürich 2, 192, 1932.Borsuk,
K. "Drei Sätze über die 
-dimensionale euklidische Sphäre." Fund. Math. 20,
177-190, 1933.Cipra, B. "If You Can't See It, Don't Believe It...."
Science 259, 26-27, 1993.Cipra, B.