At a given distance from the origin, which convex subsets of an ℓ_p-ball have the maximal volume?

Jared Irwin

Jared Irwin

Open question

2022-08-26

At a given distance from the origin, which convex subsets of an p -ball have the maximal volume?
For a positive integer n and p [ 1 , ], let B n , p := { x R n x p 1 } be the p unit-ball in R n . Fix h [ 0 , 1 ].
Of all convex subsets A of B n , p with d ( 0 , A ) = h, which shape maximizes the volume ?

Answer & Explanation

Carleigh Tate

Carleigh Tate

Beginner2022-08-27Added 8 answers

Step 1
I answered taking d ( 0 , A ) := sup a A a p . Otherwise, as pointed out in the comments by copper.hat, every set that contains the origin in its closure has zero distance.
Step 2
The shape A that minimizes the distance to the origin for a given volume is (up to sets of measure zero) a ball. This follows from the observation that if A is not (up to sets of measure zero) a ball, then the ball with radius equal to the d(0,A) has the same distance to the origin, but larger volume as d(0,A) (as A is contained in the ball of radius d(0,A)). Thus, I can shrink the radius to get the volume of A and get smaller distance to the origin. However, then A did not have minimal distance to the origin.

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