All GL (n) covariant star-body-valued valuations on convex polytopes are completely classified. It is shown that there is a unique nontrivial such valuation. This valuation turns out to be the so-called “intersection operator”—an operator that played a critical role in the solution of the Busemann-Petty problem. Introduction. A function Z defined on the set K of convex bodies (that is, of conve...

We study succinct approximation of functions that have noisy oracle access. Namely, construction of a succinct representation of a function, given oracle access to an L-approximation of the function, rather than to the function itself. Specifically, we consider the question of the succinct representation of an approximation of a convex function φ that cannot be accessed directly, but only via o...

This paper presents the concepts of (L,M)-remotehood spaces and (L,M)-convergence in framework (L,M)-fuzzy convex spaces. Firstly, it is shown that category isomorphic to Secondly, proved can be embedded as a reflective subcategory. Finally, preconvex (L,M)- convergence are introduced

Finding the minimum vertex distance between two dis-joint convex polygons in linear time. [6] Supowit, K. J.: The relative neighborhood graph, with an application to minimum spanning trees. A note on the all-nearest-neighbor problem for convex polygons. [10] O'Rourke, J., et al.: A new linear algorithm for intersecting convex polygons.Since P nw and Q R are two linearly separable convex polygon...

We prove a direct theorem for convex polynomial L p-approximation, 0 < p < 1, in terms of the classical modulus of smoothness ! 3 (f; t) p. This theorem may be regarded as an extension to L p of the well-known pointwise estimates of the Timan type and their variants of R. DeVore, Y. K. Hu and D. Leviatan for convex approximation in L p. It leads to a characterization of convex functions in Lips...

Minimizers of the total variation subject to a prescribed L-norm are considered as eigensolutions of the 1-Laplace operator. The derivation of the corresponding eigenvalue equation, which requires particular care due to the lack of smoothness, is carried out in a previous paper by using a special Lagrange multiplier rule based on Degiovanni’s weak slope. The present paper provides a simpler pro...

A language L is prefix-convex if, whenever words u and w are in L with u a prefix of w, then every word v which has u as a prefix and is a prefix of w is also in L. Similarly, we define suffix-, factor-, and subword-convex languages, where by subword we mean subsequence. Together, these languages constitute the class of convex languages which contains interesting subclasses, such as ideals, clo...

For a function defined on a convex set in a Euclidean space, midpoint convexity is the property requiring that the value of the function at the midpoint of any line segment is not greater than the average of its values at the endpoints of the line segment. Midpoint convexity is a well-known characterization of ordinary convexity under very mild assumptions. For a function defined on the integer...