Given a neighborhood graph representation of a finite set of points x i ∈ R d , i = 1, . . . , n, we construct a frame (redundant dictionary) for the space of real-valued functions defined on the graph. This frame is adapted to the underlying geometrical structure of the x i , has finitely many elements, and these elements are localized in frequency as well as in space. This construction follows the ideas of [11], with the key point that we construct a tight (or Parseval) frame. This means we have a very simple, explicit reconstruction formula for every function f defined on the graph from the coefficients given by its scalar product with the frame elements. We use this representation in the setting of denoising where we are given noisy observations of a function f defined on the graph. By applying a thresholding method to the coefficients in the reconstruction formula, we define an estimate of f whose risk satisfies a tight oracle inequality.

Objective: The insulinoma is a rare tumor of the pancreas, that can lead to spontaneous hypoglycemia due to an excessive insulin secretion. The 72-hour fast is the gold standard for finding the correct diagnosis. Endoscopic ultrasound (EUS) is an established examination method to identify the suspicious lesion. Previous studies correlate the measured size of insulinoma and their endocrine behavior. This study was designed to find a relation between these variables. Methods: We took the data of patients which had a histological confirmed insulinoma after receiving an endoscopic ultrasound in our department. Size and echogenicity were correlated with the endpoint of the 72-hour fast and hormone levels. Results: A total of 45 patients were identified. Most insulinoma were small with a volume of < 2cm3 (median 1.15cm3). There was no correlation between the duration of fasting, hormone levels and the size of the insulinoma. In addition, in a subgroup analysis, no connection could be established between the size of the insulinoma and the amount of insulin that was released after oral glucose exposure. We found out that homogeneous tumors were significantly smaller and had a lower Ki-67 index. Furthermore, there was a tendency towards a shorter period of duration for the 72-hour fast for the small tumors. Discussion: This data suggests that the measured size of insulinoma by EUS is not related to the time until termination of the 72-hour fast and measured hormone levels. The echogenicity seems more important, showing that homogenous tumors are an indicator for a higher differentiation, which can result in a shorter duration of fasting period. The differences in the secretion behavior of the insulinomas could complicate the correlation of size and duration of the 72-hour fast.

The aim of this paper is to establish two fundamental measure-metric properties of particular random geometric graphs. We consider ε-neighborhood graphs whose vertices are drawn independently and identically distributed from a common distribution defined on a regular submanifold of R K . We show that a volume doubling condition (VD) and local Poincaré inequality (LPI) hold for the random geometric graph (with high probability, and uniformly over all shortest path distance balls in a certain radius range) under suitable regularity conditions of the underlying submanifold and the sampling distribution.

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