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Better formatting, reference to Note 2 added.
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Ivan Izmestiev
Ivan Izmestiev
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The equality cases in the Brunn-Minkowski are: $A$ and $B$ lie in parallel hyperplanes (then all volumes are zero), or $A$ and $B$ are convex and homothetic.

  1. $A$ and $B$ lie in parallel hyperplanes (then all volumes are zero), or
  2. $A$ and $B$ are convex and homothetic.

A strengthening of the inequality should depend on some "non-homotheticity" measure of $A$ and $B$. There are some results in this direction, see ChapterSection 6.1 in Schneider's "Convex bodies: the Brunn-Minkowski theory", right after the proof of the BM inequality. See also Note 2 at the end of that section.

The equality cases in the Brunn-Minkowski are: $A$ and $B$ lie in parallel hyperplanes (then all volumes are zero), or $A$ and $B$ are convex and homothetic. A strengthening of the inequality should depend on some "non-homotheticity" measure of $A$ and $B$. There are some results in this direction, see Chapter 6.1 in Schneider's "Convex bodies: the Brunn-Minkowski theory", right after the proof of the BM inequality.

The equality cases in the Brunn-Minkowski are:

  1. $A$ and $B$ lie in parallel hyperplanes (then all volumes are zero), or
  2. $A$ and $B$ are convex and homothetic.

A strengthening of the inequality should depend on some "non-homotheticity" measure of $A$ and $B$. There are some results in this direction, see Section 6.1 in Schneider's "Convex bodies: the Brunn-Minkowski theory", right after the proof of the BM inequality. See also Note 2 at the end of that section.

Source Link
Ivan Izmestiev
Ivan Izmestiev
  • 6.3k
  • 26
  • 50

The equality cases in the Brunn-Minkowski are: $A$ and $B$ lie in parallel hyperplanes (then all volumes are zero), or $A$ and $B$ are convex and homothetic. A strengthening of the inequality should depend on some "non-homotheticity" measure of $A$ and $B$. There are some results in this direction, see Chapter 6.1 in Schneider's "Convex bodies: the Brunn-Minkowski theory", right after the proof of the BM inequality.