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Abstract

For compact Euclidean bodies $P,Q$, we define $\lambda(P,Q)$ to be the smallest ratio $r/s$ where $r > 0$, $s > 0$ satisfy $sQ' \subseteq P \subseteq$ $rQ''$. Here $sQ$ denotes a scaling of $Q$ by factor $s$, and $Q', Q''$ are some translates of $Q$. This function $\lambda$ gives us a new distance function between bodies wich, unlike previously studied measures, is invariant under affine transformations. If homothetic bodies are identified, the logarithm of this function is a metric. (Two bodies are {\sl homothetic} if one can be obtained from the other by scaling and translation).