3,410 research outputs found
On a strong version of the Kepler conjecture
We raise and investigate the following problem that one can regard as a very
close relative of the densest sphere packing problem. If the Euclidean 3-space
is partitioned into convex cells each containing a unit ball, how should the
shapes of the cells be designed to minimize the average surface area of the
cells? In particular, we prove that the average surface area in question is
always at least 13.8564... .Comment: 9 page
On the X-ray number of almost smooth convex bodies and of convex bodies of constant width
The X-ray numbers of some classes of convex bodies are investigated. In
particular, we give a proof of the X-ray Conjecture as well as of the
Illumination Conjecture for almost smooth convex bodies of any dimension and
for convex bodies of constant width of dimensions 3, 4, 5 and 6
From -dual sets to uniform contractions
Let denote the -dimensional Euclidean, hyperbolic, or spherical
space. The -dual set of given set in is the intersection of balls of
radii centered at the points of the given set. In this paper we prove that
for any set of given volume in the volume of the -dual set becomes
maximal if the set is a ball. As an application we prove the following. The
Kneser-Poulsen Conjecture states that if the centers of a family of
congruent balls in Euclidean -space is contracted, then the volume of the
intersection does not decrease. A uniform contraction is a contraction where
all the pairwise distances in the first set of centers are larger than all the
pairwise distances in the second set of centers. We prove the Kneser-Poulsen
conjecture for uniform contractions (with sufficiently large) in .Comment: 8 page
Isolated transfer of analog signals
Technique transfers analog signal levels across high isolation boundary without circuit performance being affected by magnetizing reactance or leakage inductance. Transfers of analog information across isolated boundary are made by interrupting signal flow, with switch, in such a manner as to produce alternating signal which is applied to transformer
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