Abstract
Letν be a signed measure on Ed with νEd=0 and ¦ν¦Ed<∞. DefineD s(ν) as sup ¦νH¦ whereH is an open halfspace. Using integral and metric geometric techniques results are proved which imply theorems such as the following.Theorem A. Letν be supported by a finite pointsetp i. ThenD s(ν)>c d(δ1/δ 2)1/2{∑ i(νp i)2}1/2 whereδ 1 is the minimum distance between two distinctp i, andδ 2 is the maximum distance. The numberc d is an absolute dimensional constant. (The number .05 can be chosen forc 2 in Theorem A.)Theorem B. LetD be a disk of unit area in the planeE 2, andp 1,p 2,...,p n be a set of points lying inD. If m if the usual area measure restricted toD, while γnP i=1/n defines an atomic measure γn, then independently of γn,nD s(m −γ n)≥ .0335n 1/4. Theorem B gives an improved solution to the Roth “disk segment problem” as described by Beck and Chen. Recent work by Beck shows thatnD s(m −γ n)≥cn 1/4(logn)−7/2.
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