The fields of Linear Programming (LP) and Geometric Measure Theory (GMT) have witnessed substantial developments since the middle of the 20th century. At the same time, they have remained mostly independent of each other. We present results on two problems related to shapes in GMT that employ results from LP, with the connections facilitated by algebraic topology. Currents represent generalized surfaces in GMT, and were introduced to study area minimizing surfaces and related problems. We consider a question of existence of certain decomposition of currents with desirable properties that has been open for decades. It turns out this question can be answered under a discrete setting using LP. We develop an analysis framework that extends this result to currents in general. In the second problem, we study a notion of average shape defined as the median of currents. In the corresponding discrete version, the median is computed efficiently using LP.