In this paper, we establish Lp norm bounds for spectral clusters on compact manifolds, under the assumption that the metric is C1,1. Precisely, we show that the Lp estimates proven by Sogge in the case of smooth metrics hold under this limited regularity assumption. It is known by examples of Smith-Sogge that such estimates fail for C1,α metrics if α < 1.

We develop faster approximation algorithms for Metric-TSP building on recent, nearly linear time approximation schemes for the LP relaxation [Chekuri and Quanrud, 2017a]. We show that the LP solution can be sparsified via cut-sparsification techniques such as those of Benczur and Karger [2015]. Given a weighted graph G with m edges and n vertices, and ǫ > 0, our randomized algorithm outputs wit...

For p ∈ [2,∞) the metric Xp inequality with sharp scaling parameter is proven here to hold true in Lp. The geometric consequences of this result include the following sharp statements about embeddings of Lq into Lp when 2 < q < p <∞: the maximal θ ∈ (0, 1] for which Lq admits a bi-θ-Hölder embedding into Lp equals q/p, and for m,n ∈ N the smallest possible bi-Lipschitz distortion of any embeddi...

Classes of convex lattice polygons which have minimal lp–perimeter with respect to the number of their vertices are said to be optimal in the sense of lp metric. The purpose of this paper is to prove the existence and explicitly find the limit shape of the sequence of these optimal convex lattice polygons as the number of their vertices tends to infinity. It is proved that if p is arbitrary int...

Intrinsic Lp metrics are defined and shown to satisfy a dimension–free bound with respect to the Hausdorff metric. MSC 2000: 52A20, 52A27, 52A40, 60G15.

Semi-metric labeling is a special case of energy minimization for pairwise Markov random fields. The energy function consists of arbitrary unary potentials, and pairwise potentials that are proportional to a given semi-metric distance function over the label set. Popular methods for solving semi-metric labeling include (i) move-making algorithms, which iteratively solve a minimum st-cut problem...

Metric labeling is a special case of energy minimization for pairwise Markov random fields. The energy function consists of arbitrary unary potentials, and pairwise potentials that are proportional to a given metric distance function over the label set. Popular methods for solving metric labeling include (i) move-making algorithms, which iteratively solve a minimum st-cut problem; and (ii) the ...

Cognitive radio (CR) systems make efficient use of the frequency spectrum by opportunistically exploiting unoccupied or under–utilized frequency bands. However, the frequency bands used by CR systems are expected to suffer from various forms of noise and interference with non–Gaussian distributions, such as narrowband and co–channel interference caused by the primary user and other CRs, respect...

We show that the Hilbert space is coarsely embeddable into any lp for 1 ≤ p < ∞. In particular, this yields new characterizations of embeddability of separable metric spaces into the Hilbert space. Coarse embeddings were defined by M. Gromov [Gr] to express the idea of inclusion in the large scale geometry of groups. G. Yu showed later that the case when a finitely generated group with a word l...