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Richard J. Gardner

  1. Operations between sets in geometry.

    Tue, 05/22/2012 - 15:01 — SomNambul
    Authors: Richard J. Gardner, Daniel Hug, Wolfgang Weil
    Subjects: Metric Geometry
    Abstract

    An investigation is launched into the fundamental characteristics of
    operations on and between sets, with a focus on compact convex sets and star
    sets (compact sets star-shaped with respect to the origin) in $n$-dimensional
    Euclidean space $\R^n$. For example, it is proved that if $n\ge 2$, with three
    trivial exceptions, an operation between origin-symmetric compact convex sets
    is continuous in the Hausdorff metric, GL(n) covariant, and associative if and
    only if it is $L_p$ addition for some $1\le p\le\infty$.

  2. Determining a rotation of a tetrahedron from a projection.

    Thu, 12/01/2011 - 15:01 — SomNambul
    Authors: Thorsten Theobald, Richard J. Gardner, Paolo Gronchi
    Subjects: Metric Geometry
    Abstract

    The following problem, arising from medical imaging, is addressed: Suppose
    that $T$ is a known tetrahedron in $\R^3$ with centroid at the origin. Also
    known is the orthogonal projection $U$ of the vertices of the image $\phi T$ of
    $T$ under an unknown rotation $\phi$ about the origin. Under what circumstances
    can $\phi$ be determined from $T$ and $U$?

  3. A problem of Klee on inner section functions of convex bodies.

    Wed, 01/19/2011 - 16:01 — SomNambul
    Authors: Richard J. Gardner, Dmitri Ryabogin, Vladyslav Yaskin, Artem Zvavitch
    Subjects: Classical Analysis and ODEs
    Abstract

    In 1969, Vic Klee asked whether a convex body is uniquely determined (up to
    translation and reflection in the origin) by its inner section function, the
    function giving for each direction the maximal area of sections of the body by
    hyperplanes orthogonal to that direction. We answer this question in the
    negative by constructing two infinitely smooth convex bodies of revolution
    about the $x_n$-axis in $\R^n$, $n\ge 3$, one origin symmetric and the other
    not centrally symmetric, with the same inner section function. Moreover, the
    pair of bodies can be arbitrarily close to the unit ball.

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