When packing convex identical 3-dimensional objects into a defined space, is a sphere the most 'efficient' shape when considering the amount of free space in the gaps?

According to the conjecture, the sphere is the convex solid which forces the largest fraction of space to remain empty.

Packed spheres leave around 25.95% of space unfilled - and intuitively would seem to be the most efficient shape.

To date, however, no mathematical proof has been discovered to show that the sphere is the most efficient.

See:Pessimal packing shapes arXiv, 2013, Metric Geometry.

Note: In 2-dimensional and 4-dimensional space, proofs have been provided for special shapes.

Also see : The Kelvin problem (3-D packing)plugin-autotooltip__plain plugin-autotooltip_bigThe Kelvin problem (3-D packing)

In 2 dimensions, the most efficient packing mechanism is an array of hexagons - a honeycomb. In 1887, William Thomson (Lord Kelvin) asked the question 'What is the most efficient 3-D packing system?“

See (the original paper) :

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