  For continuous functions, midpoint convexity characterizes convex functions.
By considering discrete versions of midpoint convexity, several types of
discrete convexities of functions, including integral convexity,
L$^\natural$-convexity and global/local discrete midpoint convexity, have been
studied. We propose a new type of discrete midpoint convexity that lies between
L$^\natural$-convexity and integral convexity and is independent of
global/local discrete midpoint convexity. The new convexity, named
DDM-convexity, has nice properties satisfied by L$^\natural$-convexity and
global/local discrete midpoint convexity. DDM-convex functions are stable under
scaling, satisfy the so-called parallelgram inequality and a proximity theorem
with the same small proximity bound as that for L$^{\natural}$-convex
functions. Several characterizations of DDM-convexity are given and algorithms
for DDM-convex function minimization are developed. We also propose
DDM-convexity in continuous variables and give proximity theorems on these
functions.
