Menu
Home
People
Places
Arts
History
Plants & Animals
Science
Life & Culture
Technology
Reference.org
Hermite constant
open-in-new
<p>In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the Hermite constant, named after <a href="/facts/Charles_Hermite/kj1bGlMr">Charles Hermite</a>, determines how long a shortest element of a <a href="/facts/Lattice_(group)/2ls9Iv9N">lattice</a> in <a href="/facts/Euclidean_space/R2UbzmzM">Euclidean space</a> can be. </p><p>The constant <i>γn</i> for integers <i>n</i> > 0 is defined as follows. For a lattice <i>L</i> in Euclidean space R<i>n</i> with unit covolume, i.e. vol(R<i>n</i>/<i>L</i>) = 1, let <i>λ</i>1(<i>L</i>) denote the least length of a nonzero element of <i>L</i>. Then √<i>γn</i> is the maximum of <i>λ</i>1(<i>L</i>) over all such lattices <i>L</i>. </p><p>The <a href="/facts/Square_root/AfrzfBdQ">square root</a> in the definition of the Hermite constant is a matter of historical convention. </p><p>Alternatively, the Hermite constant <i>γn</i> can be defined as the square of the maximal <a href="/facts/Systolic_geometry/exKrWG9e">systole</a> of a flat <i>n</i>-dimensional <a href="/facts/Torus/J9SkaV1W">torus</a> of unit volume. </p>