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Journal of Geometry, ISSN 0047-2468, 2015, Volume 106, Issue 1, pp. 63 - 73
If a convex body in K [member of] R.sup.n K a R n subtends constant visual angles over two concentric spheres exterior to K... 
52A40
Journal Article
Mathematika, ISSN 0025-5793, 01/2020, Volume 66, Issue 1, pp. 161 - 177
We prove that if K,L⊂R2 are convex bodies such that L is symmetric and the Banach–Mazur distance between K and L is equal to 2, then K is a triangle. 
52A40 (primary) | 52A10 | 52A27 (secondary)
Journal Article
Advances in Geometry, ISSN 1615-715X, 07/2019, Volume 19, Issue 3, pp. 297 - 312
A new family of ellipsoids associated with a pair of convex bodies in ℝ is introduced, which extends the classical John ellipsoids and the evolved John... 
John ellipsoid | 52A40 | Mixed volume | isotropy
Journal Article
Journal of the London Mathematical Society, ISSN 0024-6107, 06/2020, Volume 101, Issue 3, pp. 1219 - 1249
Based on reverse isoperimetric inequalities on Grassmann manifolds, a Grassmanian Loomis–Whitney inequality and its dual are established, which provides a... 
52A40 (primary)
Journal Article
Mathematika, ISSN 0025-5793, 06/2006, Volume 53, Issue 1, pp. 71 - 80
The classical Minkowski sum of convex sets is defined by the sum of the corresponding support functions. The Lp-extension of such a definition makes use of the... 
Primary, 52A40
Journal Article
Mathematika, ISSN 0025-5793, 06/2006, Volume 53, Issue 1, pp. 129 - 148
It is proved that the reciprocal of the volume of the polar bodies, about the Santaló point, of a shadow system of convex bodies Kt, is a convex function of t,... 
Primary, 52A40
Journal Article
Duke Mathematical Journal, ISSN 0012-7094, 2013, Volume 162, Issue 11, pp. 1895 - 1922
The continuity of the inverse Klain map is investigated and the class of centrally symmetric convex bodies at which every valuation depends continuously on its... 
MATHEMATICS | CENTRALLY SYMMETRIC FACES | VALUED VALUATIONS | CONVEX-BODIES | PROJECTION BODIES | MINKOWSKI-VALUATIONS | INTEGRAL GEOMETRY | MCMULLEN,P. CONJECTURE | COMPLEX VECTOR-SPACES | INTERSECTION BODIES | TRANSLATION-INVARIANT VALUATIONS | Mathematics - Metric Geometry | 52A20 | 52B45 | 52A40
Journal Article
Journal of Geometry, ISSN 0047-2468, 2015, Volume 106, Issue 1, pp. 75 - 83
In this note, we are concerned with planar convex sets whose interior is free from points of the integer lattice. Such lattice-free convex sets have been... 
52C05 | Primary 52A10 | Secondary 52A40
Journal Article
Journal of the London Mathematical Society, ISSN 0024-6107, 02/2020, Volume 101, Issue 1, pp. 393 - 431
We consider sets of trace‐normalized non‐negative operators in Hilbert–Schmidt balls that maximize their mutual Hilbert–Schmidt distance; these are optimal... 
81P45 (primary) | 52C17 (secondary) | 52A40 | 42C15 | 05B05 | MATHEMATICS | COMPLEX CODEBOOKS | MUTUALLY UNBIASED BASES | CODES | CONSTRUCTION | SYSTEMS | EQUIANGULAR TIGHT FRAMES | LINES
Journal Article
Duke Mathematical Journal, ISSN 0012-7094, 07/2010, Volume 154, Issue 1, pp. 1 - 30
A description of continuous rigid motion compatible Minkowski valuations is established. As an application, we present a Brunn-Minkowski type inequality for... 
MATHEMATICS | INEQUALITIES | INTEGRAL GEOMETRY | CONVOLUTIONS | BODIES | MULTIPLICATIVE STRUCTURE | MANIFOLDS | INVARIANT VALUATIONS | ENDOMORPHISMS | FORMULA | HARD LEFSCHETZ THEOREM | 52A90 | 43A90 | 52A40
Journal Article
Aequationes mathematicae, ISSN 0001-9054, 6/2019, Volume 93, Issue 3, pp. 619 - 628
In this note we prove the following result: for every $$\alpha \in (0,\pi )$$ α ∈ ( 0 , π ) and for a given convex body K in the plane, with minimal width w,... 
52A10 | 52A40 | Isoptic chords | Analysis | Isoptic curves | Mathematics | Euclidean disc | Combinatorics | Chords (geometry)
Journal Article
Journal of Inequalities and Applications, ISSN 1029-242X, 12/2019, Volume 2019, Issue 1, pp. 1 - 9
In the paper, we give a new generalization of Gardner–Hartenstine inequality and establish its integral form. As applications, we combine an important... 
52A40 | Hölder’s inequality | Gardner–Hartenstine’s inequality | Analysis | Mathematics, general | Mathematics | Minkowski’s inequality | Applications of Mathematics | 26D15 | Inequality
Journal Article
Journal of Functional Analysis, ISSN 0022-1236, 08/2016, Volume 271, Issue 3, pp. 610 - 619
We provide a sharp lower bound for the perimeter of the inner parallel sets of a convex body Ω. The bound depends only on the perimeter and inradius r of the... 
Convex geometry | Perimeter | Inner parallel sets | MATHEMATICS
Journal Article
Calculus of Variations and Partial Differential Equations, ISSN 0944-2669, 10/2015, Volume 54, Issue 2, pp. 1767 - 1791
We extend the classical Aleksandrov-Fenchel inequality for mixed volumes to functionals arising naturally in hermitian integral geometry. As a consequence, we... 
Primary 52A40 | Secondary 53C65 | MATHEMATICS | MATHEMATICS, APPLIED | CONVEX | THEOREM | INTEGRAL GEOMETRY | INVARIANT | BODIES | MANIFOLDS | FORMULA | Valuation
Journal Article
Mathematika, ISSN 0025-5793, 2019, Volume 65, Issue 3, pp. 500 - 504
Using previous results about shadow systems and Steiner symmetrization, we prove that the local maximizers of the volume product of convex bodies are actually... 
52A20 | 52A40 (primary) | MATHEMATICS | MATHEMATICS, APPLIED | INEQUALITIES
Journal Article
Calculus of Variations and Partial Differential Equations, ISSN 0944-2669, 4/2017, Volume 56, Issue 2, pp. 1 - 16
Recently, the duals of Federer’s curvature measures, called dual curvature measures, were discovered by Huang et al. (Acta Math 216:325–388, 2016). In the same... 
52A40 | Mathematics | Systems Theory, Control | Calculus of Variations and Optimal Control; Optimization | Analysis | Theoretical, Mathematical and Computational Physics | MATHEMATICS | MATHEMATICS, APPLIED | FIREY THEORY | VALUATIONS | SETS | AFFINE | GAUSS CURVATURE | Mathematics - Metric Geometry
Journal Article
INTERNATIONAL MATHEMATICS RESEARCH NOTICES, ISSN 1073-7928, 06/2019, Volume 2019, Issue 11, pp. 3313 - 3375
A *-linear map Phi between matrix spaces is positive if it maps positive semidefinite matrices to positive semidefinite ones, and is called completely positive... 
POLYNOMIALS | MATHEMATICS | STATES | MATRICES | OPTIMIZATION | SQUARES | SUMS
Journal Article
Duke Mathematical Journal, ISSN 0012-7094, 2014, Volume 163, Issue 11, pp. 2003 - 2022
In this note we link symplectic and convex geometry by relating two seemingly_different open conjectures: a symplectic isoperimetric-type inequality for convex... 
TOPOLOGY | MATHEMATICS | MINIMAL VOLUME-PRODUCT | SPACES | ZONOIDS | GEOMETRY | 52A20 | 52A40 | 37D50 | 52A23
Journal Article
Quaestiones mathematicae, ISSN 1607-3606, 02/2020, pp. 1 - 15
Journal Article
Advances in Mathematics, ISSN 0001-8708, 01/2017, Volume 306, pp. 51 - 88
The concept of an i-symmetrization is introduced, which provides a convenient framework for most of the familiar symmetrization processes on convex sets.... 
Convex body | Minkowski symmetrization | Central symmetrization | Schwarz symmetrization | Steiner symmetrization | MATHEMATICS | BODIES | Mathematics - Metric Geometry
Journal Article
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