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PureMathematicsnØêÆ,2023,13(6),1744-1752
PublishedOnlineJune2023inHans.https://www.hanspub.org/journal/pm
https://doi.org/10.12677/pm.2023.136178
$‘Busemann-PettyÿݯK
ÓÔ/ª
ÁÁÁkkk
Ô;ÆŒêâÆ§B²Ô;
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Á‡
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‡¥%é¡àNKÚL5`§e§‚?¿n-i-‘f˜m¤§¤n-i–‘¡NÿÝ
÷vµ(K∩ξ
⊥
)≤µ(L∩ξ
⊥
)§Ù¥ξ∈G(n,i)(1≤i≤n)§@oÙn-‘àNKÚLÿݵ(K)≤
µ(L)´Ä¤áº3®k©z¥§¼†Rubin9Zhang'u$‘Busemann-Petty¯Kn-‘
NÈ/ªƒ˜—(Ø.©y²ù‡(ØÓÔ/ª§=3þã^‡e§é?¿1≤i≤n§
kµ(K)≤n
i/2
µ(L)¤á"
'…c
ÓÔ/ª§¡N§ÿݧRadonC†
AnIsomorphicVersionoftheLower
DimensionalBusemann-Petty
ProblemsforMeasures
XianyangZhu
Scho olofDateScience,TongrenUniversity,TongrenGuizhou
©ÙÚ^:Ák.$‘Busemann-PettyÿݯKÓÔ/ª[J].nØêÆ,2023,13(6):1744-1752.
DOI:10.12677/pm.2023.136178
Ák
Received:May21
st
,2023;accepted:Jun.22
nd
,2023;published:Jun.30
th
,2023
Abstract
ThelowerdimensionalBusemann-Petty(LDBP)problemforarbitrarymeasuresasks:
ForagivenBorelmeasureµwithappropriatedensityandtwoorigin-symmetricconvex
bo diesKandL,doestheassumptionthatµ(K∩ξ
⊥
)≤µ(L∩ξ
⊥
)holdsforanyξ∈
G(n,i)(1≤i<n)implythatµ(K)≤µ(L)?Itwasprovedthattheproblemhasthesame
answerasRubinandZhang’ssolutionstotheLDBPproblemforvolumes.Inthis
paperweshowanisomorphicversionofthisresult.Namely,iftheaboveconditions
hold,thenµ(K)≤n
i/2
µ(L)forany1≤i≤n.
Keywords
IsomorphicVersion,IntersectionBody,Measures,RadonTransform
Copyright
c
2023byauthor(s)andHansPublishersInc.
ThisworkislicensedundertheCreativeCommonsAttributionInternationalLicense(CCBY4.0).
http://creativecommons.org/licenses/by/4.0/
1.Úó
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DOI:10.12677/pm.2023.1361781745nØêÆ
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DOI:10.12677/pm.2023.1361781746nØêÆ
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DOI:10.12677/pm.2023.1361781747nØêÆ
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vol
n−i
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n−i
ρ
n−i
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Z
S
n−1
ρ
i
K
(u)f(u)du=(R
n−i
f,µ)=µ(R
n−i
f),∀f∈C(S
n−1
).(9)
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ρ
i
K
=R
t
n−i
µ.
^ ÎÒI
n
n−i
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n
n−i
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n
n−i
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ω
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i
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n−1
α(t)dt−ω
Z
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n−i−1
α(t)dt≤
ω
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i
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n−1
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t
n−i−1
α(t)dt,(10)
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+
¥,‡Œê:8,α(t)>0¤á,@
o…=a=bž(10)ª´˜‡ðª.
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a
i
Z
b
0
t
n−i−1
α(t)dt−a
i
Z
a
0
t
n−i−1
α(t)dt≤
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n−1
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t
n−1
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•Ò´
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
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t

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t
n−1
α(t)dt≤
Z
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a
t
n−1
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´a>b,Øª(11)w,¤á.
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a
0
t
n−1
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i
Z
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n−i−1
α(t)dt=
Z
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0
t
n−1
α(t)dt−a
i
Z
b
0
t
n−i−1
α(t)dt.
DOI:10.12677/pm.2023.1361781748nØêÆ
Ák
=´
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t
n−1
α(t)dt+a
i
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0
t
n−1
α(t)dt+a
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Z
a
0
t
n−i−1
α(t)dt,
du¼ê
R
s
0
t
n−1
α(t)dtÚa
i
R
s
0
t
n−i−1
α(t)dt3s>0žüN5(Ï•šK¼êα(t)Øt∈R
+
¥,
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n
¥ü‡:é¡àN,K½Â§‚ƒmBanach-Mazurål•
d
BM
(K,L)=inf{d>0:∃T∈GL(n):K⊂TL⊂dK},
…P
d
I
K=min{d
BM
(K,L):L•¡N}.
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1,loc
(R
n
)´˜‡óšKëY¼ê,µ´—ݼê•fk•BorelÿÝ,
…K,L⊂R
n
´ü‡:é¡àN,÷v^‡
µ(K∩ξ
⊥
)≤µ(L∩ξ
⊥
),ξ∈G(n,i),1≤i≤n−1,(12)
@ok
µ(K)≤d
i
I
(K)µ(L).
y²Šâ(7)ª-(12)•
R
n−i

Z
ρ
K
(·)
0
f(t·)t
n−i−1
dt

(ξ
⊥
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n−i

Z
ρ
L
(·)
0
f(t·)t
n−i−1
dt

(ξ
⊥
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I
(K)Q(du ¡N‚5C†”E,´˜‡¡N)
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n
,ëY/|^Ún2.2,KQ∈I
n
n−i
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u(n−i)-¡NQ),|^(6),(9),
Z
S
n−1
ρ
i
Q
(u)
Z
ρ
K
(u)
0
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S
n−1
ρ
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(u)
Z
ρ
L
(u)
0
f(tu)t
n−i−1
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(u),a=ρ
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(u),b=ρ
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ρ
i
Q
(u)
ρ
i
K
(u)
Z
ρ
K
(u)
0
t
n−1
f(tu)dt−ρ
i
Q
(u)
Z
ρ
K
(u)
0
t
n−i−1
f(tu)dt
≤
ρ
i
Q
(u)
ρ
i
K
(u)
Z
ρ
L
(u)
0
t
n−1
f(tu)dt−ρ
i
Q
(u)
Z
ρ
L
(u)
0
t
n−i−1
f(tu)dt(15)
é(15)3S
n−1
þÈ©,•â(14)ÚQ⊂K⊂d
I
(K)Q,·‚¼
Z
S
n−1
ρ
i
Q
(u)
ρ
i
K
(u)
Z
ρ
K
(u)
0
t
n−1
f(tu)dtdu≤
Z
S
n−1
ρ
i
Q
(u)
ρ
i
K
(u)
Z
ρ
L
(u)
0
t
n−1
f(tu)dtdu,
DOI:10.12677/pm.2023.1361781749nØêÆ
Ák
=
1
d
i
I
(K)
Z
S
n−1
Z
ρ
K
(u)
0
t
n−1
f(tu)dtdu≤
Z
S
n−1
Z
ρ
L
(u)
0
t
n−1
f(tu)dtdu.(16)
ù‡ØªÙ¢Ò´eª4‹I/ª
µ(K)≤d
i
I
(K)µ(L).
w,/,î¼ü ¥B
n
2
´‡¡N,•âJohn½n,9é?¿:é¡àNK⊂R
n
kd
BM
(K,B
n
2
)≤
√
n¤á¯¢,†½n2.4˜Œ±ïXeíØ.
íØ2.5f∈L
1,loc
(R
n
)´˜óšKëY¼ê,µ´k—ݼê•fk•BorelÿÝ,
KÚL´R
n
¥ü‡:é¡àN§XJ÷veª
µ(K∩ξ
⊥
)≤µ(L∩ξ
⊥
),ξ∈G(n,i),1≤i≤n−1,
@ok
µ(K)≤n
i/2
µ(L).
5µ½n2.4ÚíØ2.5¥i=1œ/3©z[3]¥®²y².
Ä7‘8
I[g,‰ÆÄ7]Ï‘8(10801140)¶Ô;ÆƬ‰ïéÄÄ7‘8(trxyDH2220)"
ë•©z
[1]Zvavitch,A.(2005)TheBusemann-PettyProblemforArbitraryMeasures.Mathematische
Annalen,331,867-887.https://doi.org/10.1007/s00208-004-0611-5
[2]Koldobsky,A.(2000)AFunctionalAnalyticApproachtoIntersectionBodies.Geometricand
FunctionalAnalysis,10,1507-1526.https://doi.org/10.1007/PL00001659
[3]Koldobsky,A.andZvavitch,A.(2015)AnIsomorphicVersionoftheBusemann-PettyProblem
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