Intersection of ball with hyperplane

[dawo]

Hello Everyone!

I have the following problem: Consider the scalled unit sphere w.r.t. the p-norm, i.e. tB with

B = {x in R^n: ||x||_p = 1}, t>0 and p>1

and the hyperplane

H = {x in R^n: sum x_i = 1}.

I am interested in the intersection of H and tB (of course only in the cases where it is not empty). My conjecture is that all the points in the intersection have the same distance to (1/n, ..., 1/n) (in the p-norm).

My questions:
(1) Is this correct?
(2) If yes, what is the distance?

I would be very thankful for either a solution or a hint where to look this up.

regards
dawo