Characterization of ellipses as uniformly dense sets with respect to a family of convex bodies

Let K \subset R^N be a convex body containing the origin. A measurable set G \subset R^N with positive Lebesgue measure is said to be uniformly K-dense if, for any fixed r > 0, the measure of G \cap (x + rK) is constant when x varies on the boundary of G (here, x + rK denotes a translation of a dilation of K). We first prove that G must always be strictly convex and at least C1,1-regular; also, if K is centrally symmetric, K must be strictly convex, C1,1-regular and such that K = G - G up to homotheties; this implies in turn that G must be C2,1- regular. Then for N = 2, we prove that G is uniformly K-dense if and only if K and G are homothetic to the same ellipse. This result was already proven by Amar, Berrone and Gianni in [3]. However, our proof removes their regularity assumptions on K and G and, more importantly, it is susceptible to be generalized to higher dimension since, by the use of Minkowski's inequality and an affine inequality, avoids the delicate computations of the higher-order terms in the Taylor expansion near r = 0 for the measure of G\cap(x+rK) (needed in [3])

The heart of a convex body

We investigate some basic properties of the {\it heart} $\heartsuit(\mathcal{K})$ of a convex set $\mathcal{K}.$ It is a subset of $\mathcal{K},$ whose definition is based on mirror reflections of euclidean space, and is a non-local object. The main motivation of our interest for $\heartsuit(\mathcal{K})$ is that this gives an estimate of the location of the hot spot in a convex heat conductor with boundary temperature grounded at zero. Here, we investigate on the relation between $\heartsuit(\mathcal{K})$ and the mirror symmetries of $\mathcal{K};$ we show that $\heartsuit(\mathcal{K})$ contains many (geometrically and phisically) relevant points of $\mathcal{K};$ we prove a simple geometrical lower estimate for the diameter of $\heartsuit(\mathcal{K});$ we also prove an upper estimate for the area of $\heartsuit(\mathcal{K}),$ when $\mathcal{K}$ is a triangle.Comment: 15 pages, 3 figures. appears as "Geometric Properties for Parabolic and Elliptic PDE's", Springer INdAM Series Volume 2, 2013, pp 49-6

The False Claims Act\u27s First-to-File Bar: Jurisdictional or Not?

While the general theory behind the First-to-File bar may appear relatively simple, properly and practically applying it is a difficult task. Though there are many complexities involved with First-to-File litigation, this Article focuses on the current disagreement among the various circuit courts of appeals as to whether the First-to-File bar is a jurisdictional bar to litigation. This Article is not intended to offer an exhaustive analysis or resolution to this issue, but rather, will simply introduce the current debate and offer initial thoughts on why the authors believe the First-to-File bar is a non-jurisdictional provision

Row orientation effects on whole-canopy gas exchange of potted and field-grown grapevines

The effects of canopy orientation (North-South vs. East-West) on total canopy assimilation (TCA) and transpiration (TCE) were evaluated on potted grapevines mounted on wheeled platforms for full swivel relation. Eight vines were assembled in pairs to form four canopy walls 2 m long, 1.1 m tall and 0.25-0.30 m wide. TCA and TCE readings were also taken in the field on four NS-oriented, hedgerow cordon-trained grapevines. Diurnal trends of TCA recorded on potted vines showed little variability when related to row orientation. The TCE pattern for EW followed essentially that of light intensity, whereas a NS orientation induced a marked decrease in TCE at midday before recovering in mid-afternoon. As a result, water use efficiency (WUE) in NS rows was higher during the midday hours. Total canopy water loss in NS was linearly correlated with estimates of intercepted light, suggesting that water use was a function of both, light intensity and canopy geometry (i.e. more light lost to the ground at noon, hence less transpiration). The results for the NS-oriented field-grown canopies differed to some extent from those of the pot experiments. TCA showed a more marked afternoon decline and TCE flattened at noon, though with no apparent decrease. WUE efficiency was lowest at the highest evaporative demand. The daily water loss of field vines could not be predicted by total light interception estimates only, indicating a more complex regulation of canopy transpiration than recorded on potted plants