  In response to subtle adversarial examples flipping classifications of neural
network models, recent research has promoted certified robustness as a
solution. There, invariance of predictions to all norm-bounded attacks is
achieved through randomised smoothing of network inputs. Today's
state-of-the-art certifications make optimal use of the class output scores at
the input instance under test: no better radius of certification (under the
$L_2$ norm) is possible given only these score. However, it is an open question
as to whether such lower bounds can be improved using local information around
the instance under test. In this work, we demonstrate how today's "optimal"
certificates can be improved by exploiting both the transitivity of
certifications, and the geometry of the input space, giving rise to what we
term Geometrically-Informed Certified Robustness. By considering the smallest
distance to points on the boundary of a set of certifications this approach
improves certifications for more than $80\%$ of Tiny-Imagenet instances,
yielding an on average $5 \%$ increase in the associated certification. When
incorporating training time processes that enhance the certified radius, our
technique shows even more promising results, with a uniform $4$ percentage
point increase in the achieved certified radius.
