At a given distance from the origin, which convex subsets of an ℓ 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 ℓ 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 ?

Question

At a given distance from the origin, which convex subsets of an       ℓ    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       ℓ    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 ?

Carleigh Tate · Accepted Answer

Step 1I 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 2The 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.

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

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