The Bregman distance
B_{\xi_x}(y,x)
B
ξ
x
(
y
,
x
)
,
\xi_x \in \partial J(y),
ξ
x
∈
∂
J
(
y
)
,
associated to a convex sub-differentiable functional
J
J
is known to be in general non-symmetric in its arguments
x
x
,
y
y
. In this note we address the question when Bregman distances can be bounded against each other when the arguments are switched, i.e., if some constant
C>0
C
>
0
exists such that for all
x,y
x
,
y
on a convex set
M
M
it holds that
\frac{1}{C} B_{\xi_x}(y,x) \leq B_{\xi_y}(x,y) \leq C B_{\xi_x}(y,x).
1
C
B
ξ
x
(
y
,
x
)
≤
B
ξ
y
(
x
,
y
)
≤
C
B
ξ
x
(
y
,
x
)
.
We state sufficient conditions for such an inequality and prove in particular that it holds for the
p
p
-powers of the
\ell_p
ℓ
p
and
L^p
L
p
-norms when
1 < p <\infty
1
<
p
<
∞
.