Content-based image authentication typically assesses authenticity based on a distance measure between the image to be tested and its original. Commonly employed distance measures such as the Minkowski measures (including Hamming and Euclidean distances) may not be adequate for content-based image authentication since they do not exploit statistical and spatial properties in features. This pape...

The Brunn Minkowski theory of convex bodies and mixed volumes has provided many tools for solving problems involving projections and valuations of compact convex sets in Euclidean space. Among the most beautiful results of twentieth century convexity is Hadwiger's characterization theorem for the elementary mixed volumes (Quermassintegrals); (see [3, 5, 9]). Hadwiger's characterization leads to...

Let d be a positive integer and U ⊂ ℤd finite. We study $$\beta (U): = \mathop {\inf }\limits_{\mathop {A,B \ne \phi }\limits_{{\rm{finite}}} } {{\left| {A + B U} \right|} \over {{{\left| A \right|}^{1/2}}{{\left| \right|}^{1/2}}}},$$ other related quantities. employ tensorization, which is not available for the doubling constant, ∣U U∣/∣U∣. For instance, we show (U) \left| \right|,$$ whenever ...