TY - JOUR
T1 - Convex geometry and waist inequalities
AU - Klartag, Bo’az
N1 - Publisher Copyright:
© 2017, Springer International Publishing.
PY - 2017/2/1
Y1 - 2017/2/1
N2 - This paper presents connections between Gromov’s work on isoperimetry of waists and Milman’s work on the M-ellipsoid of a convex body. It is proven that any convex body K⊆ Rn has a linear image K~ ⊆ Rn of volume one satisfying the following waist inequality: Any continuous map f: K~ → Rℓ has a fiber f- 1(t) whose (n- ℓ) -dimensional volume is at least cn - ℓ, where c> 0 is a universal constant. In the specific case where K= [ 0 , 1 ] n it is shown that one may take K~ = K and c= 1 , confirming a conjecture by Guth. We furthermore exhibit relations between waist inequalities and various geometric characteristics of the convex body K.
AB - This paper presents connections between Gromov’s work on isoperimetry of waists and Milman’s work on the M-ellipsoid of a convex body. It is proven that any convex body K⊆ Rn has a linear image K~ ⊆ Rn of volume one satisfying the following waist inequality: Any continuous map f: K~ → Rℓ has a fiber f- 1(t) whose (n- ℓ) -dimensional volume is at least cn - ℓ, where c> 0 is a universal constant. In the specific case where K= [ 0 , 1 ] n it is shown that one may take K~ = K and c= 1 , confirming a conjecture by Guth. We furthermore exhibit relations between waist inequalities and various geometric characteristics of the convex body K.
UR - http://www.scopus.com/inward/record.url?scp=85011838380&partnerID=8YFLogxK
U2 - 10.1007/s00039-017-0397-8
DO - 10.1007/s00039-017-0397-8
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AN - SCOPUS:85011838380
SN - 1016-443X
VL - 27
SP - 130
EP - 164
JO - Geometric and Functional Analysis
JF - Geometric and Functional Analysis
IS - 1
ER -