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Is there an English alternative to Blaschke's "Kreis und Kugel"? (As you may guess, "Kreis und Kugel" was not translated.)

In other words I am looking for a well written and very elementary book which talks about

  • Hausdorff convergence and compactness,
  • Isoperimetric inequality, Brunn–Minkowski,
  • and so on.
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    $\begingroup$ Can you clarify how elementary a book you have in mind? E.g., do you expect a reader to know measure theory and point-set topology? (It's been a long time since I looked at Kreis und Kugel and I don't recall its level.) $\endgroup$
    – Mark Meckes
    Commented Aug 25, 2011 at 18:54
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    $\begingroup$ However there is a Russian translation: bookshunt.ru/b149794_krug_i_shar $\endgroup$
    – Anatoly Kochubei
    Commented Aug 25, 2011 at 19:00
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    $\begingroup$ @Mark; imagine a "good undergraduate student"; i.e., a person who is ready to think but knows nothing. $\endgroup$
    – Anton Petrunin
    Commented Aug 25, 2011 at 19:05

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This seems such an obvious suggestion that I hesitate to make it, but I believe it meets all the criteria you state: Rolf Schneider's Convex Bodies: The Brunn-Minkowski Inequality.

If this isn't what you're after, maybe you could clarify your conditions?

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  • $\begingroup$ I suspect this is less elementary than Anton has in mind. (I'm guessing that Anton is not looking for something for himself to read here.) $\endgroup$
    – Mark Meckes
    Commented Aug 25, 2011 at 18:51
  • $\begingroup$ That is at least something --- thank you. Although I believe this book will scare some students (the style is a bit formal). $\endgroup$
    – Anton Petrunin
    Commented Aug 25, 2011 at 18:54
  • $\begingroup$ Ah, yes. I was thinking that it's elementary in the sense of assuming little background, but I can see that it might be rather forbidding. $\endgroup$
    – Tom Leinster
    Commented Aug 25, 2011 at 23:40
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Keith Ball's "An elementary introduction to modern convex geometry" (which is a chapter in Flavors of Geometry) is very well-written and elementary, although the content may be different from what you have in mind. It's very good, for example, on the Brunn-Minkowski and isoperimetric inequalities, but I'm not sure it mentions Hausdorff convergence at all.

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  • $\begingroup$ No; I mean there is no Hausdorff convergence in there. $\endgroup$
    – Anton Petrunin
    Commented Aug 25, 2011 at 19:32
  • $\begingroup$ I suspect that finding an "elementary" book on convex geometry that contains Hausdorff convergence is rather unlikely, since the notion of convergence pretty much requires at least one course in real analysis. My suggestion is to use a book like Ball or Klain-Rota and discuss other topics that require Hausdorff convergence as additional topics. $\endgroup$
    – Deane Yang
    Commented Aug 25, 2011 at 20:15
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    $\begingroup$ Maybe it makes sense to translate "Kreis und Kugel"... $\endgroup$
    – Anton Petrunin
    Commented Aug 25, 2011 at 20:34

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