  We introduce a new notion of influence for symmetric convex sets over
Gaussian space, which we term "convex influence". We show that this new notion
of influence shares many of the familiar properties of influences of variables
for monotone Boolean functions $f: \{\pm1\}^n \to \{\pm1\}.$
  Our main results for convex influences give Gaussian space analogues of many
important results on influences for monotone Boolean functions. These include
(robust) characterizations of extremal functions, the Poincar\'e inequality,
the Kahn-Kalai-Linial theorem, a sharp threshold theorem of Kalai, a stability
version of the Kruskal-Katona theorem due to O'Donnell and Wimmer, and some
partial results towards a Gaussian space analogue of Friedgut's junta theorem.
The proofs of our results for convex influences use very different techniques
than the analogous proofs for Boolean influences over $\{\pm1\}^n$. Taken as a
whole, our results extend the emerging analogy between symmetric convex sets in
Gaussian space and monotone Boolean functions from $\{\pm1\}^n$ to $\{\pm1\}$
