Let
$X$
be a vector space and let
$\unicode[STIX]{x1D711}:X\rightarrow \mathbb{R}\cup \{-\infty ,+\infty \}$
be an extended real-valued function. For every function
$f:X\rightarrow \mathbb{R}\cup \{-\infty ,+\infty \}$
, let us define the
$\unicode[STIX]{x1D711}$
-envelope of
$f$
by
$$\begin{eqnarray}f^{\unicode[STIX]{x1D711}}(x)=\sup _{y\in X}\unicode[STIX]{x1D711}(x-y)\begin{array}{@{}c@{}}-\\ \cdot \end{array}f(y),\end{eqnarray}$$
where
$\begin{array}{@{}c@{}}-\\ \\ \\ \cdot \end{array}$
denotes the lower subtraction in
$\mathbb{R}\cup \{-\infty ,+\infty \}$
. The main purpose of this paper is to study in great detail the properties of the important generalized conjugation map
$f\mapsto f^{\unicode[STIX]{x1D711}}$
. When the function
$\unicode[STIX]{x1D711}$
is closed and convex,
$\unicode[STIX]{x1D711}$
-envelopes can be expressed as Legendre–Fenchel conjugates. By particularizing with
$\unicode[STIX]{x1D711}=(1/p\unicode[STIX]{x1D706})\Vert \cdot \Vert ^{p}$
, for
$\unicode[STIX]{x1D706}>0$
and
$p\geqslant 1$
, this allows us to derive new expressions of the Klee envelopes with index
$\unicode[STIX]{x1D706}$
and power
$p$
. Links between
$\unicode[STIX]{x1D711}$
-envelopes and Legendre–Fenchel conjugates are also explored when
$-\unicode[STIX]{x1D711}$
is closed and convex. The case of Moreau envelopes is examined as a particular case. In addition to the
$\unicode[STIX]{x1D711}$
-envelopes of functions, a parallel notion of envelope is introduced for subsets of
$X$
. Given subsets
$\unicode[STIX]{x1D6EC}$
,
$C\subset X$
, we define the
$\unicode[STIX]{x1D6EC}$
-envelope of
$C$
as
$C^{\unicode[STIX]{x1D6EC}}=\bigcap _{x\in C}(x+\unicode[STIX]{x1D6EC})$
. Connections between the transform
$C\mapsto C^{\unicode[STIX]{x1D6EC}}$
and the aforestated
$\unicode[STIX]{x1D711}$
-conjugation are investigated.