The inverse Kakeya problem

@article{Cabello2022TheIK,
  title={The inverse Kakeya problem},
  author={Sergio Cabello and Otfried Cheong and Michael Gene Dobbins},
  journal={Periodica Mathematica Hungarica},
  year={2022},
  volume={84},
  pages={70-75}
}
We prove that the largest convex shape that can be placed inside a given convex shape  $$Q \subset \mathbb {R}^{d}$$ Q ⊂ R d in any desired orientation is the largest inscribed ball of  Q . The statement is true both when “largest” means “largest volume” and when it means “largest surface area”. The ball is the unique solution, except when maximizing the perimeter in the two-dimensional case. 

References

SHOWING 1-10 OF 13 REFERENCES
The reverse Kakeya problem
Abstract We prove a generalization of Pál's conjecture from 1921: if a convex shape P can be placed in any orientation inside a convex shape Q in the plane, then P can also be turned continuously
Recent work connected with the Kakeya problem
where Sn−1 is the unit sphere in R. This paper will be mainly concerned with the following issue, which is still poorly understood: what metric restrictions does the property (1) put on the set E?
Geometry and Convexity: A Study in Mathematical Methods
Helps students see mathematics as an organic whole by focusing on the geometric while presenting viewpoints and methods that require a general understanding and unification of previous mathematical
From rotating needles to stability of waves; emerging connections between combinatorics, analysis and PDE
We survey the interconnections between geometric combinatorics (such as the Kakeya problem), arithmetic combinatorics (such as the classical problem of determining which sets contain arithmetic
From harmonic analysis to arithmetic combinatorics
Arithmetic combinatorics, or additive combinatorics, is a fast developing area of research combining elements of number theory, combinatorics, harmonic analysis and ergodic theory. Its arguably
Some problems on maxima and minima regarding ovals
  • The Science Report of the Tohoku Imperial University,
  • 1917
Convex Surfaces. Interscience tracts in pure and applied mathematics
  • Interscience Publishers,
  • 1958
...
...