Geometric Tomography: Parallel X-rays of planar convex bodies

By Richard J. Gardner. 2006

850 citations recorded by the source corpus.

View Paper: https://www.semanticscholar.org/paper/b59c8810e63685a7d6d766119834478561859df3

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Citations in the corpus

The even Orlicz-Minkowski problem (2010, background): “Good general references for the theory of convex bodies are provided by the books of Gardner [11], Gruber [14], Schneider [59], and Thompson [64].”
General $L_p$ affine isoperimetric inequalities (2008, background): “[11] R. J. Gardner and G. Zhang, Affine inequalities and radial mean bodies Amer.”
On the L dual Minkowski problem (2018, background): “1 tackling the famous Busemann-Petty problem, see Gardner [21], Gardner, Koldobsky & Schlumprecht [23], Koldobsky [38–40], Lutwak [48], and Zhang [67].”
The dual Minkowski problem for symmetric convex bodies (2017, background): “[18] R.J. Gardner, The Brunn-Minkowski inequality, Bull.”
The dual Orlicz–Brunn–Minkowski theory ☆ (2014, unknown): “The dual Brunn-Minkowski theory has connections and applications to integral geometry, Minkowski geometry, the local theory of Banach spaces, geometric tomography, and stereology; see [13] and the references given there.”
Valuations and Busemann{Petty Type Problems (2008, background): “Although the notion of intersection bodies is relatively new, the topic has been intensively studied in recent years, see [17,20,27,34,40,51] and the books [13,35,65].”
OPERATIONS BETWEEN SETS IN GEOMETRY (2012, background): “It also has connections and applications to integral geometry, Minkowski geometry, the local theory of Banach spaces, and stereology; see [16] and the references given there.”
Moment-entropy inequalities (2004, background): “See, for example, the books of Gardner (1995), Leichtweiß (1998), Schneider (1993) and Thompson (1996) [and also the article of Bourgain and Milman (1987)] for references regarding the Blaschke–Santaló inequality.”
Remarks on the conjectured log-Brunn–Minkowski inequality (2013, background): “We refer to [16], [19], [21] for details and references.”
GL(n) equivariant Minkowski valuations (2013, background): “The operator Π is a GL(n) contravariant continuous Minkowski valuation, see for example [10].”
Rotation equivariant Minkowski valuations (2012, background): “In recent years it has attracted increased attention due to its numerous applications in different areas, see [3, 4, 5, 7, 13, 31].”
Volume inequalities for asymmetric Wulff shapes (2011, background): “As a general reference, the reader may wish to consult the books [13, 37] and the articles [25, 26].”
The Busemann-Petty problem for arbitrary measures (2004, background): “Minkowski’s theorem (see [Ga3]) shows that an origin-symmetric starshaped body is uniquely determined by the volume of its hyperplane sections (the same is true for the case of general symmetric measure, see Corollary 1 below).”
Orlicz–John ellipsoids (2014, background): “Very recently, one essential obstacle in the development of Orlicz Brunn–Minkowski theory, what is the lack of a notion corresponding to Lp addition, has been smoothed by Gardner, Hug and Weil [14,15].”
Symmetrization in geometry (2016, background): “There are many other symmetrization processes in geometry, such as those leading to the fundamental notions of projection body, intersection body, and centroid body (see [8, 28]).”