TY - JOUR

T1 - Errata

T2 - Corrigendum to “Complemented Brunn–Minkowski inequalities and isoperimetry for homogeneous and non-homogeneous measures” (Adv. Math. 262 (2014) (867–908) S0001870814002072 (10.1016/j.aim.2014.05.023))

AU - Milman, Emanuel

AU - Rotem, Liran

PY - 2017/2/5

Y1 - 2017/2/5

N2 - Page 879, Corollary 2.17 Replace the sentence “Let [Formula Presented] denote two Borel sets with [Formula Presented]” with “Let [Formula Presented] denote two convex sets with [Formula Presented]”. Also erase the last sentence (“Furthermore, if b=0 then...”) which is now redundant. The proof of the corollary remains exactly the same. Convexity of A and B was implicitly assumed, as it is needed in order to assert that Ψ is concave. Consequently, the sentence in the Introduction on Page 870 “.. it is not difficult to show that up to sets of zero μ-measure, homothetic (and possibly translated) copies of K are the unique isoperimetric minimizers”, should be modified to “.. are the unique convex isoperimetric minimizers”. Page 889, Lemma 4.8 The statement of the Lemma is correct as written. However, the first paragraph of the proof again implicitly assumes that A and B are convex. To handle the general case, erase the first paragraph completely and replace the second paragraph with: “By the homogeneity of μ, we may scale B to a set [Formula Presented] so that [Formula Presented], and set [Formula Presented]. Employing the homogeneity and using the assumption, we have: [Formula Presented] The proof continues as written, using weights δ, 1 − δ instead of [Formula Presented]. Page 891, Theorem 4.10 Again, the proof of the Theorem implicitly assumes that B and [Formula Presented] are convex. However, the theorem remains true under the much weaker assumption that B is star-shaped, and C is an arbitrary Borel set with [Formula Presented]. This follows immediately from the more general Theorem 5.15, by choosing [Formula Presented] (see also Remark 5.9 for the case of non-absolutely continuous measure μ).

AB - Page 879, Corollary 2.17 Replace the sentence “Let [Formula Presented] denote two Borel sets with [Formula Presented]” with “Let [Formula Presented] denote two convex sets with [Formula Presented]”. Also erase the last sentence (“Furthermore, if b=0 then...”) which is now redundant. The proof of the corollary remains exactly the same. Convexity of A and B was implicitly assumed, as it is needed in order to assert that Ψ is concave. Consequently, the sentence in the Introduction on Page 870 “.. it is not difficult to show that up to sets of zero μ-measure, homothetic (and possibly translated) copies of K are the unique isoperimetric minimizers”, should be modified to “.. are the unique convex isoperimetric minimizers”. Page 889, Lemma 4.8 The statement of the Lemma is correct as written. However, the first paragraph of the proof again implicitly assumes that A and B are convex. To handle the general case, erase the first paragraph completely and replace the second paragraph with: “By the homogeneity of μ, we may scale B to a set [Formula Presented] so that [Formula Presented], and set [Formula Presented]. Employing the homogeneity and using the assumption, we have: [Formula Presented] The proof continues as written, using weights δ, 1 − δ instead of [Formula Presented]. Page 891, Theorem 4.10 Again, the proof of the Theorem implicitly assumes that B and [Formula Presented] are convex. However, the theorem remains true under the much weaker assumption that B is star-shaped, and C is an arbitrary Borel set with [Formula Presented]. This follows immediately from the more general Theorem 5.15, by choosing [Formula Presented] (see also Remark 5.9 for the case of non-absolutely continuous measure μ).

KW - Borell–Brascamp–Lieb inequality

KW - Complemented Brunn–Minkowski inequality

KW - Corrigendum

KW - Homogeneous measures

KW - Isoperimetry on cones

UR - http://www.scopus.com/inward/record.url?scp=84994121001&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84994121001&partnerID=8YFLogxK

U2 - 10.1016/j.aim.2016.08.017

DO - 10.1016/j.aim.2016.08.017

M3 - Comment/debate

AN - SCOPUS:84994121001

VL - 307

SP - 1378

EP - 1379

JO - Advances in Mathematics

JF - Advances in Mathematics

SN - 0001-8708

ER -