Convexity in the widest sense (e.g., also with its combinatorial, discrete, and stochastic aspects) is an interesting research field in the heart of mathematics, since many mathematical disciplines (like optimization, convex analysis, functional analysis, computer science (via computational geometry), discrete mathematics, combinatorics,...) need basic notions, tools, and methods from there. The field is mainly geometric in nature; therefore also some basic books on general classical geometry contain chapters/parts referring to convexity (see, e.g., [100, Chapter 12] and [156]). The refreshing character of the field mainly comes from the fact that one has immediate approach to the studied objects (e.g., strictly convex bodies, smooth convex bodies, convex polytopes, etc.), which are at first glance not too complicated and geometrically defined. (Strongly enriching the geometric part, in Minkowski geometry a second convex body, the unit ball, comes additionally into the game.) But regarding the used methods the field is very wide; these methods can come from analysis, stochastic, discrete geometry, combinatorics, integral and differential geometry, classical geometry, etc.! In other words, we have a clearly defined, geometrically inspiring family of studied objects, and these bring together mathematicians from very different fields to look at “the same geometric objects” with very different viewpoints (a property that, although being even “more narrow”, also constant width sets have, see Chapter 1 above). An additional, but related reason for the refreshing beauty of convexity is given by the richness of fascinating geometric properties that already certain special types of convex bodies have. We mention some of them, in all cases also citing basic surveys or books presenting these families in a broad manner. Namely, we have, for example, *simplices* (see [1040] and [527, Section 2], *classes of polytopes with different degrees of symmetry*, like uniform polytopes, semiregular polytopes, or isogonal ones (see, e.g., [540, § 23], [268], [986, pp. 378–382], [100, § 12.5], [595], [784], [812], and [527, § 1.1]), *balls* (cf. [540, § 32], [160, § 16], [125], [100, Chapters 10, 18, and 20], [419], and Subsection 1.2 in [527]), *ellipsoids* (cf. [160, § 70], [932], [478], [100, Chapters 14 and 15], [527, § 1.3]), and [82], *zonoids and zonotopes* (here we refer to the excellent surveys [1043] and [444]), and the three closely related classes *sets of constant width* (see [238], [527, Section 5] and, for Minkowski spaces, [793, Section 2]), *complete sets* (seemingly no comprehensive survey is existent, but the topic is also discussed in [238], [527], and [793]), and *reduced sets* (cf. [701] and, for normed spaces, [702]). It would be nice to have a monograph on all these (and more) classes of convex sets listing their interesting properties and applications, because they are useful and essential in many directions. For example, they can be important as special unit balls of Banach spaces, as inspiring objects to understand phenomena in high dimensions, as projection bodies (zonoids and, in the polytopal case, zonotopes centered at the origin), and so on.

The history of convexity started in ancient times, with problems in the spirit of the isoperimetric question, and connected with famous geometers like Euclid, Archimedes, and others. Later, the growing theoretical building of this field was enriched by contributions of Kepler, Euler, Cauchy, Steiner, Brunn, Minkowski, Blaschke, Busemann, Eggleston, Hadwiger, A. D. Aleksandrov, Coxeter, L. Fejes Tóth, Klee, Grünbaum, Schneider as well as many other excellent mathematicians (it is impossible to write a complete list of really important contributors). Regarding detailed representations of the history of convexity we refer to the articles [347], [475], [627], and [473], recommendable refreshing expository articles on convex sets in general are [471], [101], [633], and [70]. A very important, influential book that gave directions for further research, fundamental and also responsible for the way that convexity has chosen over the 20th century, was [160]. In this monograph the field itself was represented and summarized for the first time. Later everything was growing faster, and leading experts tried to structurize and organize the field by books like the Proceedings to the 1965 Colloquium on Convexity in Copenhagen (Kobenhavns Univ. Mat. Inst., Copenhagen, 1967), [630], [1129], and [482], which are collections of important surveys and articles on different aspects and subfields of convexity. Then, in 1993, even a Handbook of Convex Geometry (see [483]) appeared, filled again with many excellent surveys which cover the most important partial fields of convexity. We mention that there are also other Handbooks (on Banach spaces, or on discrete and computational geometry, etc.) containing a lot of material closely related to convexity. And also the books of the Hungarian series usually named “Intuitive Geometry” contain very many contributions to convexity. More modern milestones are the already mentioned monographs [1039], [401], [461], [1124], and [477]. Further books which illuminate special aspects of convexity or have textbook character were already mentioned in the Notes of Chapter 1, and we shortly repeat them here: early examples are [132] and [160], later continued by [1204], [312], [737], [1140], [98] (containing also a chapter on Minkowski geometry), [737], [706], [618], [1134], [654], [1163], and [719]. And we also have to mention the problem books [629], [311], [272], [635], and [182]. In particular, the problem book [272] contains (after its preface) a useful list of further related problem collections and standard references from convexity. This list should be extended by books on general convexity which are not mentioned in Chapter 1 (since bodies of constant width do not occur in these books). These are [15], [205], [502], [503], [493], [711] (also partially considering Minkowski spaces), [82], [714], and [1080]. Clearly, in all these books various aspects and properties of convex sets are discussed with different focal points. Since our book contains also aspects of discrete and combinatorial geometry of convex bodies and polytopes (see, e.g., Chapters 6 and 15), we mention here also books going into these more special directions. Regarding polytope theory, there are the basic monographs [16], [489], [813], [188], and [1223], whereas the books [505], [151], and [152] refer to the combinatorial geometry of convex bodies. Furthermore, [345], [981], [479], [163], [801], and [113] are important in discrete geometry and related fields. Analytic aspects of convex sets, widely presented in [461] and [464], are also discussed in the more recent important publications [649], [650], and [992]. Referring to finite-dimensional real Banach spaces in the spirit of our Chapter 10 and the monograph [1124], we also mention the books [938], [186], and [41]; several survey articles in the Handbook [596] are also closely related. And for all these aspects, we again refer to further excellent surveys in the Handbook [483]. The most comprehensive and also deepest monograph referring to the central topics of the field is [1039].

There are also geometrical fields that, at least some decades ago, had almost no direct connections to convex geometry. As non-specialist, one could guess that *algebraic geometry* is one of them. However, the monographs [897] and [325] clearly show that there are many deep cross-connections between both fields, e.g., referring to polytope theory, mixed volumes, Ehrhart polynomials, and further topics. See, for example, Section 12.4, and also the excellently grown field of *tropical geometry* should be mentioned here.

Finally we mention here at least one well-known example for applications of convexity. Convex analysis (see the classical book [978] and also [546]) originally belonged to pure mathematics. But roughly since 1960, applied aspects started to play an increasingly important role, also in connection with non-smooth analysis (since convexity allows the usage of a differential calculus going further than the classical one). Due to this, convex analysis became more important for optimization, approximation theory, inverse problems, etc.; a suitable related reference still close to classical convexity in our sense here is [1100].

**Ellipsoids**

Besides bodies of constant width, ellipsoids represent a very important class of special convex bodies. Therefore also here (like in Section 2.12) they deserve to be discussed separately. Ellipsoids have many captivating properties and have been studied for their own intrinsic interest. Many extremal problems of affine nature have ellipsoids as extremal bodies, and perhaps this is the reason for the extensive research. In Minkowski geometry, Hilbert geometry, and affine differential geometry, ellipsoids play an important role, see also Subsection 11.6. A good example is the already mentioned article [206]. All ten problems posed there (only one of them being solved) refer to possible ellipsoid characterizations in Euclidean space, but the motivations come from Minkowski geometry. Surveys on ellipsoids in convex geometry are the papers of [932], [478], and Section 3 in [527].

The first proof of the False Center Theorem can be found in [685], although the proof presented here is in [852]. Concerning characterizations of ellipsoids by means of their sections or projections there has been a considerable amount of research. Before Larman’s paper, among others, there is work by Rogers [980] and Aitchison [5]. After Larman’s paper, there is a series of publications by Burton [200], Burton and Larman [201], Burton and Mani [202] and Larman, Montejano and Morales [686], where an interesting theory has been developed. See also the related work of Gruber and Ódor [480]. Here we also mention a nice result of Burton [199] generalizing a theorem of Aitchison: a convex body of dimension \(n > 2\) all whose hyperplane sections cutting from it a piece of sufficiently small volume are centrally symmetric, is the Minkowski sum of an ellipsoid and a zonotope. Further important characterizations of ellipsoids were obtained in [469] and in [1079].

The uniqueness of the largest ellipsoid in a convex body and the smallest ellipsoid containing it was first proved in the 2-dimensional case by Behrend [95]. The case of uniqueness of the smallest ellipsoid containing a given convex set was deduced by John [594] from general results on extremum problems with inequalities as constraints. The name Löwner ellipsoid was first used by Busemann [203], but refers to the ellipsoid with given center and smallest volume containing a convex body, see Theorem 2.12.2. Busemann mentioned that Löwner did not publish his result. Proofs for the Löwner and John ellipsoid were given by Danzer, Laugwitz and Lenz [279]. Gruber [472] proved that most convex bodies in \(\mathbb {E}^n\), in the sense of the Baire category, touch the boundaries of their John and the Löwner ellipsoid in precisely \(n(n+3)/2\) points.