On the Volume of the Intersection of Two L n p Balls

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Abstract: This note deals with the following problem, the case

p = 1

,

q = 2

of which was introduced to us by Vitali Milman: What is the volume left in the

L_{p}^{n}

ball after removing a t-multiple of the

L_{q}^{n}

ball? Recall that the

L_{r}^{n}

ball is the set

and note that for

0 < p < q < i n f t y

the

L_{q}^{n}

ball is contained in the

L_{p}^{n}

ball. In Corollary 4 we show that, after normalizing Lebesgue measure so that the volume of the

L_{p}^{n}

ball is one, the answer to the problem above is of order

e^{- c t^{p} n^{p / q}}

for

T < t < 1 o v e r 2 n^{1 o v e r p - 1 o v e r q}

, where

c

and

T

depend on

p

and

q

but not on

n

. The main theorem, Theorem 3, deals with the corresponding question for the surface measure of the

L_{p}^{n}

sphere.

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