Property of A Centrally Symmetric Polyhedron in 3-space
There is a geometric inequality that is in a sense dual to the isoperimetric inequality in the following sense. Both involve a length and an area. The isoperimetric inequality is an upper bound for area in terms of length. There is a geometric inequality which provides an upper bound for a certain length in terms of area. More precisely it can be described as follows.
Any centrally symmetric convex body of surface area can be squeezed through a noose of length, with the tightest fit achieved by a sphere. This property is equivalent to a special case of Pu's inequality, one of the earliest systolic inequalities.
For example, an ellipsoid is an example of a convex centrally symmetric body in 3-space. It may be helpful to the reader to develop an intuition for the property mentioned above in the context of thinking about ellipsoidal examples.
An alternative formulation is as follows. Every convex centrally symmetric body in admits a pair of opposite (antipodal) points and a path of length joining them and lying on the boundary of, satisfying
Read more about this topic: Introduction To Systolic Geometry
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