  Given a measurable set $A\subset \mathbb R^d$ we consider the "large-distance
graph" $\mathcal{G}_A$, on the ground set $A$, in which each pair of points
from $A$ whose distance is bigger than 2 forms an edge. We consider the
problems of maximizing the $2d$-dimensional Lebesgue measure of the edge set as
well as the $d$-dimensional Lebesgue measure of the vertex set of a
large-distance graph in the $d$-dimensional Euclidean space that contains no
copies of a complete graph on $k$ vertices. The former problem may be seen as a
continuous analogue of Tur\'an's classical graph theorem, and the latter as a
graph-theoretic analogue of the classical isodiametric problem. Our main result
yields an analogue of Mantel's theorem for large-distance graphs. Our approach
employs an isodiametric inequality in an annulus, which might be of independent
interest.
