Motivated by the inequality
,
Carbery (2009) raised the question of what is the “right” analogue of this estimate in
for
.
Carlen, Frank, Ivanisvili and Lieb (2018) recently obtained an
version of this inequality by providing upper bounds for
in terms of
the quantities
,
and
when
, and lower
bounds when
,
thereby proving (and improving) the suggested possible inequalities of
Carbery. We continue investigation in this direction by refining the
estimates of Carlen, Frank, Ivanisvili and Lieb. We obtain upper bounds for
also when
and lower
bounds when
.
For
we extend our upper bounds to any finite number of functions.
In addition, we show that all our upper and lower bounds of
for
,
, are the best possible in
terms of the quantities
,
and
, and
we characterize the equality cases.
Keywords
triangle inequality, $L^p$ spaces, concave envelopes,
Bellman function
