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The setup is: consider a shape $S\in R^d$, and let $\mu_S$ be uniform measure on $S$. I want to find (among all shape with volume 1) the shape that minimizes $\mathbb{E}_{X,Y \sim \mu_S i.i.d}[\|X-Y\|^k]$.

I came across this problem in one step of my final project for a class. Intuition tells me it would be a ball since the ball is a convex shape and doesn't have sharp corners, but I just cannot come up with a rigorous proof. It has a isoperimetric inequality flavor... I think someone must have looked into this problem before, I wonder can anyone point me to some potential resources? Thanks a lot!

Xiaoyun
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  • Please don't only put the question in the title, the text of the question must be understandable on it's own. Do you mean two points on the surface? And what distance are you talking about? The standard euclidian distance in the room where the shape(s) live or the distances along the surface of the shape, or ... – Henrik supports the community Dec 19 '24 at 14:20
  • https://math.stackexchange.com/questions/2694329/shape-with-area-a-and-shortest-average-distance-between-any-two-points?rq=1 This question covers the 2D case. Maybe that can help? – Toph Dec 19 '24 at 14:28
  • Thanks so much! The 2D case helps. Yeah I should be more clear in the setup. I meant two points sampled inside the shape, not just on the surface. And I am considering just Euclidean distance. I will modify the setup and maybe post another one. – Xiaoyun Dec 20 '24 at 06:49

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