On the Brunn-Minkowski theorem

✍ Scribed by H. Groemer

• Book ID: 104653321
• Publisher: Springer
• Year: 1988
• Tongue: English
• Weight: 568 KB
• Volume: 27
• Category: Article
• ISSN: 0046-5755
• DOI: 10.1007/bf00181500

✦ Synopsis

Let K 1 and K 2 be n-dimensional convex bodies. If V denotes volume the Brunn-Minkowski theorem in its simplest form states that V(K 1 + K2) 1/"/> V(Ka) ~1" + V(K2) TM, and that equality holds if and only if K a and K 2 are homothetic. We consider the following associated stability problem: If V(K 1 + K2) 1/" differs not more than e from V(Ka) 1/" + V(K2) ~/" how close is Kx (in terms of the Hausdorff metric) to a nearest homothetic copy of K27 Several theorems that answer questions of this kind are proved. These results can also 1~ expressed as inequalities that are stronger than the original Brunn-Minkowski inequality. Furthermore, some consequences concerning stability statements for other geometric inequalities are discussed.