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).
- Borell–Brascamp–Lieb inequality
- Complemented Brunn–Minkowski inequality
- Homogeneous measures
- Isoperimetry on cones