Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: New bounds on the average distance from the Fermat-Weber center of a
  planar convex body   The Fermat-Weber center of a planar body $Q$ is a point in the plane from
which the average distance to the points in $Q$ is minimal. We first show that
for any convex body $Q$ in the plane, the average distance from the
Fermat-Weber center of $Q$ to the points of $Q$ is larger than ${1/6} \cdot
\Delta(Q)$, where $\Delta(Q)$ is the diameter of $Q$. This proves a conjecture
of Carmi, Har-Peled and Katz. From the other direction, we prove that the same
average distance is at most $\frac{2(4-\sqrt3)}{13} \cdot \Delta(Q) < 0.3490
\cdot \Delta(Q)$. The new bound substantially improves the previous bound of
$\frac{2}{3 \sqrt3} \cdot \Delta(Q) \approx 0.3849 \cdot \Delta(Q)$ due to
Abu-Affash and Katz, and brings us closer to the conjectured value of ${1/3}
\cdot \Delta(Q)$. We also confirm the upper bound conjecture for centrally
symmetric planar convex bodies.

22
The categories of this paper are: chao-dyn
11
The categories of this paper are: hep-lat
26
The categories of this paper are: bayes-an
28
The categories of this paper are: alg-geom
23
The categories of this paper are: solv-int
4
The categories of this paper are: cond-mat
6
The categories of this paper are: astro-ph
21
The categories of this paper are: adap-org
1
The categories of this paper are: cs
29
The categories of this paper are: funct-an
24
The categories of this paper are: dg-ga
33
The categories of this paper are: plasm-ph
34
The categories of this paper are: chem-ph
10
The categories of this paper are: q-bio
18
The categories of this paper are: econ
19
The categories of this paper are: eess