Convex figures and polyhedra by L. A. Lyusternik (PDF)

Description

x + 176 pages including index, elementary discussion of polyhedral figures, bw illustrations

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐I found this book interesting because in really brought back the geometry to the subject (i.e. proofs in the style of Euclid but with “modern” terminology and considerations). Also, the types of theorems covered in here are “historic” and interesting, it really feels like the author is presenting the proof great mathematics such as Cauchy and Minkowski gave just with modern language and considerations (not necessarily modern terminology though). And while it was published roughly 50 years ago and is an English translation, I felt like it could have been written today and would be understandable. With all the good praise I give the book, I know that there are some things that others might not find as enjoyable: the author relies, admittedly, on intuition and some diagrams to work through a proof and there are no exercises. Further, a lot of terms are not rigorously defined at the on set and are left for the last 2 sections which give all the “formal” definitions and terminology (e.g. epsilon-delta definitions of continuity and limit, compactness, etc.) but I feel that if the reader has some taken some analysis, they would be able to recognize when the author is using ideas such as neighborhoods and converging sequences and be able to translate said ideas if wanted. Also, since there are no exercises in the book this means that if your taking a class in convex geometry ,then it can only be used as a supplement. Furthermore, I feel like I should make a point of saying that most of the theorems are proved in dimensions 2 and 3 with a usual comments that the proofs are exactly the same for dimensions 4 and higher; I like this for it gives the reader the opportunity to “finish” a proof for the arbitrary n-dimensional case and feel a slight sense of satisfaction. While some readers might not like this type of writing (motivation first formalism 2nd), I did personally not find this to be such a big problem for I had already taken classes such as real analysis and partook in an independent study in convex geometry using a modern book.With this said, I would like to think I had a good grasp of the general concepts of the subject from a modern point of view before I picked up this book. I hoped this Convex Figures would satisfy my craving for more of the geometry that is sometimes missed nowadays in the language of real analysis (e.g. interior points of the intersections of compact n-dimensional convex bodies which are affinely dependent). And I believe it did. I am happy with my purchase and I strongly recommend Convex Figures and Polyhedron to anyone with a strong interest and craving for the geometry that goes into convex geometry.

⭐This book has a lot of geometric information about the topic of its title, and is written at a level that is accessible (mostly) to people with only high school geometry. (One chapter requires calculus.) The primary flaws in the book are not the author’s, but the translator’s. I am sure that Lyusternik referred to the items in question by their normal Russian terminology, but the translator in some cases seems not to be familiar with enough mathematical terminology (I assume, from his name, that English is his first language, so it’s not weakness in English that is his problem). For example, he uses the term “prismoid” for an antiprism, refers to “ovals of constant width” (the normal term would be “curves of constant width”), and there are other such errors.So I would advise the reader not to accept unquestioned the terminology presented in this book, though the geometrical insight is excellent. This is one reason to dock the book a star from the five it might otherwise get.

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