Geometry Conjecture

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2 total messages Started by Brian Sutin Thu, 05 Nov 1992 00:31

#101

Author: Brian Sutin
Date: Thu, 05 Nov 1992 00:31

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I need the following result (which may be common knowledge):

Let D be a region of R^3 with a reasonable boundary S and let dS be any
measure on the surface of D.  If

    INT(S){ 1/|x - y| dS(y) } = 1	for all x in D,

then D must be an R^3 ball.

Brian Sutin				sutin@lick.ucsc.edu
Lick Observatory, UCSC			Santa Cruz, CA 95064
    Fred:   "May I rescue you?"
    Ginger: "No, thank you.  I prefer being in distress."

Re: Geometry Conjecture

#104

Author: Paul R. Chernoff
Date: Fri, 06 Nov 1992 21:23

24 lines
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In article <1d9q1sINN9ms@darkstar.UCSC.EDU> sutin@helios.ucsc.edu (Brian Sutin) writes:
>I need the following result (which may be common knowledge):
>
>Let D be a region of R^3 with a reasonable boundary S and let dS be any
>measure on the surface of D.  If
>
>    INT(S){ 1/|x - y| dS(y) } = 1	for all x in D,
>
>then D must be an R^3 ball.
>

This is certainly false if dS is allowed to be *any* measure.  For
let S be any nice closed surface in R^3. Think of S as a conductor and
let dS represent an equilibrium distribution of positive charge on S;
then the field inside S is 0 and the potential is constant.  (For the case
of an ellipsoid, where it is shown explicitly that dS is a positive measure,
see Kellogg, Foundations of Potential Theory, pp 188-191.)

-- 
# Paul R. Chernoff                       chernoff@math.berkeley.edu     #
# Department of Mathematics              ucbvax!math!chernoff           #
# University of California               chernoff%math@ucbvax.bitnet    #
# Berkeley, CA  94720                                                   #

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