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The Princeton Companion to Mathematics by Imre Leader, June Barrow-Green, Timothy Gowers

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IV.26 High-Dimensional Geometry and Its Probabilistic Analogues

Keith Ball

1 Introduction

If you have ever watched a child blowing soap bubbles, then you cannot have failed to notice that the bubbles are, at least as far as the human eye can tell, perfectly spherical. From a mathematical perspective, the reason for this is simple. The surface tension in the soap solution causes each bubble to make its area as small as possible, subject to the constraint that it encloses a fixed amount of air (and cannot compress the air too much). The sphere is the surface of smallest area that encloses a given volume.

As a mathematical principle, this seems to have been recognized by the ancient Greeks, although fully rigorous demonstrations did not appear ...

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